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== 7.4#h(0.6em)Factor Special Products

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "open parenthesis 12 x close parenthesis squared .")[$attach(( 12 x ), t: 2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "144 x squared")[$144 x^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis m plus 4 close parenthesis squared .")[$attach(( m + 4 ), t: 2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "m squared plus 8 m plus 16")[$m^(2) + 8 m + 16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis p minus 9 close parenthesis squared .")[$attach(( p − 9 ), t: 2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "p squared minus 18 p plus 81")[$p^(2) − 18 p + 81$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis k plus 3 close parenthesis open parenthesis k minus 3 close parenthesis .")[$( k + 3 ) ( k − 3 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "k squared minus 9")[$k^(2) − 9$]
]
]

The strategy for factoring we developed in the last section will guide you as you factor most binomials, trinomials, and polynomials with more than three terms. We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.


=== Factor Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.

#figure(figph[This image shows the FOIL procedure for multiplying (3x + 4) squared. The polynomial is written with two factors (3x + 4)(3x + 4). Then, the terms are 9 x squared + 12 x + 12 x + 16, demonstrating first, outer, inner, last. Finally, the product is written, 9 x squared + 24 x + 16.], alt: "This image shows the FOIL procedure for multiplying (3x + 4) squared. The polynomial is written with two factors (3x + 4)(3x + 4). Then, the terms are 9 x squared + 12 x + 12 x + 16, demonstrating first, outer, inner, last. Finally, the product is written, 9 x squared + 24 x + 16.", caption: none)

The first term is the square of the first term of the binomial and the last term is the square of the last. The middle term is twice the product of the two terms of the binomial.

#math.equation(block: true, alt: "open parenthesis 3 x close parenthesis squared plus 2 open parenthesis 3 x times 4 close parenthesis plus 4 squared; 9 x squared plus 24 x plus 16")[$attach(( 3 x ), t: 2) + 2 ( 3 x · 4 ) + 4^(2) \ 9 x^(2) + 24 x + 16$]The trinomial 9#emph[x]#super[2] + 24#emph[x] +16 is called a perfect square trinomial. It is the square of the binomial 3#emph[x]+4.

We’ll repeat the Binomial Squares Pattern here to use as a reference in factoring.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Binomial Squares Pattern]

If #emph[a] and #emph[b] are real numbers,

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared, open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) &  &  & #h(2em) attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2)$]
]

When you square a binomial, the product is a perfect square trinomial. In this chapter, you are learning to factor—now, you will start with a perfect square trinomial and factor it into its prime factors.

You could factor this trinomial using the methods described in the last section, since it is of the form #emph[ax]#super[2] + #emph[bx] + #emph[c]. But if you recognize that the first and last terms are squares and the trinomial fits the #strong[perfect square trinomials pattern], you will save yourself a lot of work.

Here is the pattern—the reverse of the binomial squares pattern.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Perfect Square Trinomials Pattern]

If #emph[a] and #emph[b] are real numbers,

#math.equation(block: true, alt: "a squared plus 2 a b plus b squared equals open parenthesis a plus b close parenthesis squared, a squared minus 2 a b plus b squared equals open parenthesis a minus b close parenthesis squared")[$a^(2) + 2 a b + b^(2) = attach(( a + b ), t: 2) &  &  & #h(2em) a^(2) − 2 a b + b^(2) = attach(( a − b ), t: 2)$]
]

To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, #math.equation(block: false, alt: "a squared")[$a^(2)$]. Next check that the last term is a perfect square, #math.equation(block: false, alt: "b squared")[$b^(2)$]. Then check the middle term—is it twice the product, 2#emph[ab]? If everything checks, you can easily write the factors.

#examplebox("Example 1")[How to Factor Perfect Square Trinomials][
Factor: #math.equation(block: false, alt: "9 x squared plus 12 x plus 4")[$9 x^(2) + 12 x + 4$].

#solutionbox[
#figure(figph[This table gives the steps for factoring 9 x squared +12 x +4. The first step is recognizing the perfect square pattern “a” squared + 2 a b + b squared. This includes, is the first term a perfect square and is the last term a perfect square. The first term can be written as (3 x) squared and the last term can be written as 2 squared. Also, in the first step, the middle term has to be twice “a” times b. This is verified by 2 times 3 x times 2 being 12 x.], alt: "This table gives the steps for factoring 9 x squared +12 x +4. The first step is recognizing the perfect square pattern “a” squared + 2 a b + b squared. This includes, is the first term a perfect square and is the last term a perfect square. The first term can be written as (3 x) squared and the last term can be written as 2 squared. Also, in the first step, the middle term has to be twice “a” times b. This is verified by 2 times 3 x times 2 being 12 x.", caption: none)

#figure(figph[The second step is writing the square of the binomial. The polynomial is written as (3 x) squared + 2 times 3 x times 2 + 2 squared. This is factored as (3 x + 2) squared.], alt: "The second step is writing the square of the binomial. The polynomial is written as (3 x) squared + 2 times 3 x times 2 + 2 squared. This is factored as (3 x + 2) squared.", caption: none)

#figure(figph[The last step is to check with multiplication.], alt: "The last step is to check with multiplication.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "4 x squared plus 12 x plus 9")[$4 x^(2) + 12 x + 9$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 x plus 3 close parenthesis squared")[$attach(( 2 x + 3 ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "9 y squared plus 24 y plus 16")[$9 y^(2) + 24 y + 16$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 y plus 4 close parenthesis squared")[$attach(( 3 y + 4 ), t: 2)$]
]
]

The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern #math.equation(block: false, alt: "a squared minus 2 a b plus b squared")[$a^(2) − 2 a b + b^(2)$], which factors to #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared")[$attach(( a − b ), t: 2)$].

The steps are summarized here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor perfect square trinomials.]

#math.equation(block: true, alt: "Step 1. Does the trinomial fit the pattern?, a squared plus 2 a b plus b squared, a squared minus 2 a b plus b squared; times Is the first term a perfect square?, open parenthesis a close parenthesis squared, open parenthesis a close parenthesis squared; Write it as a square.; times Is the last term a perfect square?, open parenthesis a close parenthesis squared open parenthesis b close parenthesis squared, open parenthesis a close parenthesis squared open parenthesis b close parenthesis squared; Write it as a square.; times Check the middle term. Is it 2 a b ?, open parenthesis a close parenthesis squared ↘ 2 times a times b ↙ open parenthesis b close parenthesis squared, open parenthesis a close parenthesis squared ↘ 2 times a times b ↙ open parenthesis b close parenthesis squared; Step")[$"Step 1." #h(0.2em) "Does the trinomial fit the pattern?" &  &  & a^(2) + 2 a b + b^(2) &  &  & #h(2em) a^(2) − 2 a b + b^(2) \ #h(2.5em) ⋅ #h(0.5em) "Is the first term a perfect square?" &  &  & attach(( a ), t: 2) &  &  & #h(2em) attach(( a ), t: 2) \ #h(4em) "Write it as a square." &  &  &  &  &  &  \ #h(2.5em) ⋅ #h(0.5em) "Is the last term a perfect square?" &  &  & attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) &  &  & #h(2em) attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) \ #h(4em) "Write it as a square." &  &  &  &  &  &  \ #h(2.5em) ⋅ #h(0.5em) "Check the middle term. Is it" #h(0.2em) 2 a b ? &  &  & attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) &  &  & #h(2em) attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) \ "Step 2." #h(0.2em) "Write the square of the binomial." &  &  & attach(( a + b ), t: 2) &  &  & #h(2em) attach(( a − b ), t: 2) \ "Step 3." #h(0.2em) "Check by multiplying." &  &  &  &  &  &$]
]

We’ll work one now where the middle term is negative.

#examplebox("Example 2")[][
Factor: #math.equation(block: false, alt: "81 y squared minus 72 y plus 16")[$81 y^(2) − 72 y + 16$].

#solutionbox[
The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared")[$attach(( a − b ), t: 2)$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression 81y^2 - 72y + 16 is displayed on a white background.], alt: "A mathematical expression 81y^2 - 72y + 16 is displayed on a white background.", caption: none)]),
  [Are the first and last terms perfect squares?], [#figure(figph[Two mathematical expressions are shown: (9y)^2 and (4)^2, displayed on a white background.], alt: "Two mathematical expressions are shown: (9y)^2 and (4)^2, displayed on a white background.", caption: none)],
  [Check the middle term.], [#figure(figph[Calculation showing the middle term 2(9y)(4), which simplifies to 72y, derived from the squared terms (9y)^2 and (4)^2 in a binomial expansion.], alt: "Calculation showing the middle term 2(9y)(4), which simplifies to 72y, derived from the squared terms (9y)^2 and (4)^2 in a binomial expansion.", caption: none)],
  [Does it match #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared")[$attach(( a − b ), t: 2)$]? Yes.], [#figure(figph[The algebraic expression (9y)^2 - 2 \* 9y \* 4 + 4^2, illustrating the perfect square trinomial identity a^2 - 2ab + b^2, with 'a' as 9y and 'b' as 4.], alt: "The algebraic expression (9y)^2 - 2 * 9y * 4 + 4^2, illustrating the perfect square trinomial identity a^2 - 2ab + b^2, with 'a' as 9y and 'b' as 4.", caption: none)],
  [Write the square of a binomial.], [#figure(figph[A mathematical expression shows the quantity (9y - 4) squared. The expression is enclosed in parentheses, with '9y' followed by a minus sign, then '4', and finally a superscript '2' outside the closing parenthesis.], alt: "A mathematical expression shows the quantity (9y - 4) squared. The expression is enclosed in parentheses, with '9y' followed by a minus sign, then '4', and finally a superscript '2' outside the closing parenthesis.", caption: none)],
  [Check by multiplying.], [],
  [#math.equation(block: false, alt: "open parenthesis 9 y minus 4 close parenthesis squared")[$attach(( 9 y − 4 ), t: 2)$]], [],
  [#math.equation(block: false, alt: "open parenthesis 9 y close parenthesis squared minus 2 times 9 y times 4 plus 4 squared")[$attach(( 9 y ), t: 2) − 2 ⋅ 9 y ⋅ 4 + 4^(2)$]], [],
  [#math.equation(block: false, alt: "81 y squared minus 72 y plus 16 ✓")[$81 y^(2) − 72 y + 16 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "64 y squared minus 80 y plus 25")[$64 y^(2) − 80 y + 25$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 8 y minus 5 close parenthesis squared")[$attach(( 8 y − 5 ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "16 z squared minus 72 z plus 81")[$16 z^(2) − 72 z + 81$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 z minus 9 close parenthesis squared")[$attach(( 4 z − 9 ), t: 2)$]
]
]

The next example will be a perfect square trinomial with two variables.

#examplebox("Example 3")[][
Factor: #math.equation(block: false, alt: "36 x squared plus 84 x y plus 49 y squared")[$36 x^(2) + 84 x y + 49 y^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the mathematical expression 36x^2 + 84xy + 49y^2.], alt: "The image displays the mathematical expression 36x^2 + 84xy + 49y^2.", caption: none)]),
  [Test each term to verify the pattern.], [#figure(figph[An algebraic expression demonstrating the perfect square trinomial identity a^2 + 2ab + b^2, shown as (6x)^2 + 2 \* 6x \* 7y + (7y)^2.], alt: "An algebraic expression demonstrating the perfect square trinomial identity a^2 + 2ab + b^2, shown as (6x)^2 + 2 * 6x * 7y + (7y)^2.", caption: none)],
  [Factor.], [#figure(figph[A mathematical expression displays a binomial in parentheses, (6x + 7y), raised to the power of 2, indicating the square of the sum of two terms.], alt: "A mathematical expression displays a binomial in parentheses, (6x + 7y), raised to the power of 2, indicating the square of the sum of two terms.", caption: none)],
  [Check by multiplying.], [],
  [#math.equation(block: false, alt: "open parenthesis 6 x plus 7 y close parenthesis squared")[$attach(( 6 x + 7 y ), t: 2)$]], [],
  [#math.equation(block: false, alt: "open parenthesis 6 x close parenthesis squared plus 2 times 6 x times 7 y plus open parenthesis 7 y close parenthesis squared")[$attach(( 6 x ), t: 2) + 2 ⋅ 6 x ⋅ 7 y + attach(( 7 y ), t: 2)$]], [],
  [#math.equation(block: false, alt: "36 x squared plus 84 x y plus 49 y squared ✓")[$36 x^(2) + 84 x y + 49 y^(2) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "49 x squared plus 84 x y plus 36 y squared")[$49 x^(2) + 84 x y + 36 y^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 7 x plus 6 y close parenthesis squared")[$attach(( 7 x + 6 y ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "64 m squared plus 112 m n plus 49 n squared")[$64 m^(2) + 112 m n + 49 n^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 8 m plus 7 n close parenthesis squared")[$attach(( 8 m + 7 n ), t: 2)$]
]
]

#examplebox("Example 4")[][
Factor: #math.equation(block: false, alt: "9 x squared plus 50 x plus 25")[$9 x^(2) + 50 x + 25$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "9 x squared plus 50 x plus 25")[$9 x^(2) + 50 x + 25$]]),
  [Are the first and last terms perfect squares?], [#math.equation(block: false, alt: "open parenthesis 3 x close parenthesis squared open parenthesis 5 close parenthesis squared")[$attach(( 3 x ), t: 2) #h(3em) attach(( 5 ), t: 2)$]],
  [Check the middle term—is it #math.equation(block: false, alt: "2 a b ?")[$2 a b ?$]], [#math.equation(block: false, alt: "open parenthesis 3 x close parenthesis squared ↘ 2 open parenthesis 3 x close parenthesis open parenthesis 5 close parenthesis 30 x ↙ open parenthesis 5 close parenthesis squared")[$attach(( 3 x ), t: 2) "↘" limits(zws)_(limits(2 ( 3 x ) ( 5 ))_(30 x)) "↙" attach(( 5 ), t: 2)$]],
  [No! #math.equation(block: false, alt: "30 x not equal to 50 x")[$30 x ≠ 50 x$]], [This does not fit the pattern!],
  [Factor using the “ac” method.], [#math.equation(block: false, alt: "9 x squared plus 50 x plus 25")[$9 x^(2) + 50 x + 25$]],
  [#math.equation(block: false, alt: "Notice: a c; 9 times 25; 225 and 5 times 45 equals 225; 5 plus 45 equals 50")[$"Notice:" #h(0.2em) a c \ 9 · 25 \ 225 #h(0.5em) "and" #h(0.5em) 5 · 45 = 225 \ 5 + 45 = 50$]], [],
  [Split the middle term. #linebreak() Factor by grouping.], [#math.equation(block: false, alt: "9 x squared plus 5 x plus 45 x plus 25; x open parenthesis 9 x plus 5 close parenthesis plus 5 open parenthesis 9 x plus 5 close parenthesis; open parenthesis 9 x plus 5 close parenthesis open parenthesis x plus 5 close parenthesis")[$9 x^(2) + 5 x + 45 x + 25 \ x ( 9 x + 5 ) + 5 ( 9 x + 5 ) \ ( 9 x + 5 ) ( x + 5 )$]],
  [Check. #linebreak() #math.equation(block: false, alt: "open parenthesis 9 x plus 5 close parenthesis open parenthesis x plus 5 close parenthesis; 9 x squared plus 45 x plus 5 x plus 25; 9 x squared plus 50 x plus 25 ✓")[$#h(2.5em) ( 9 x + 5 ) ( x + 5 ) \ #h(2.5em) 9 x^(2) + 45 x + 5 x + 25 \ #h(2.5em) 9 x^(2) + 50 x + 25 #h(0.2em) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "16 r squared plus 30 r s plus 9 s squared")[$16 r^(2) + 30 r s + 9 s^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 8 r plus 3 s close parenthesis open parenthesis 2 r plus 3 s close parenthesis")[$( 8 r + 3 s ) ( 2 r + 3 s )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "9 u squared plus 87 u plus 100")[$9 u^(2) + 87 u + 100$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 u plus 4 close parenthesis open parenthesis 3 u plus 25 close parenthesis")[$( 3 u + 4 ) ( 3 u + 25 )$]
]
]

Remember the very first step in our Strategy for Factoring Polynomials? It was to ask “is there a greatest common factor?” and, if there was, you factor the GCF before going any further. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.

#examplebox("Example 5")[][
Factor: #math.equation(block: false, alt: "36 x squared y minus 48 x y plus 16 y")[$36 x^(2) y − 48 x y + 16 y$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "36 x squared y minus 48 x y plus 16 y")[$36 x^(2) y − 48 x y + 16 y$]]),
  [Is there a GCF? Yes, 4#emph[y], so factor it out.], [#math.equation(block: false, alt: "4 y open parenthesis 9 x squared minus 12 x plus 4 close parenthesis")[$4 y ( 9 x^(2) − 12 x + 4 )$]],
  [Is this a perfect square trinomial?], [],
  [Verify the pattern.], [#figure(figph[A mathematical expression showing the expansion of a perfect square trinomial, (3x - 2)^2, within brackets, multiplied by 4y, with the general form a^2 - 2ab + b^2 highlighted in red above.], alt: "A mathematical expression showing the expansion of a perfect square trinomial, (3x - 2)^2, within brackets, multiplied by 4y, with the general form a^2 - 2ab + b^2 highlighted in red above.", caption: none)],
  [Factor.], [#math.equation(block: false, alt: "4 y open parenthesis 3 x minus 2 close parenthesis squared")[$4 y attach(( 3 x − 2 ), t: 2)$]],
  [Remember: Keep the factor 4#emph[y] in the final product.], [],
  [Check.], [],
  [#math.equation(block: false, alt: "4 y open parenthesis 3 x minus 2 close parenthesis squared")[$4 y attach(( 3 x − 2 ), t: 2)$]], [],
  [#math.equation(block: false, alt: "4 y open bracket open parenthesis 3 x close parenthesis squared minus 2 times 3 x times 2 plus 2 squared close bracket")[$4 y [ attach(( 3 x ), t: 2) − 2 · 3 x · 2 + 2^(2) ]$]], [],
  [#math.equation(block: false, alt: "4 y open parenthesis 9 x squared minus 12 x plus 4 close parenthesis")[$4 y ( 9 x^(2) − 12 x + 4 )$]], [],
  [#math.equation(block: false, alt: "36 x squared y minus 48 x y plus 16 y ✓")[$36 x^(2) y − 48 x y + 16 y ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "8 x squared y minus 24 x y plus 18 y")[$8 x^(2) y − 24 x y + 18 y$].

#solutionbox[
#math.equation(block: true, alt: "2 y open parenthesis 2 x minus 3 close parenthesis squared")[$2 y attach(( 2 x − 3 ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "27 p squared q plus 90 p q plus 75 q")[$27 p^(2) q + 90 p q + 75 q$].

#solutionbox[
#math.equation(block: true, alt: "3 q open parenthesis 3 p plus 5 close parenthesis squared")[$3 q attach(( 3 p + 5 ), t: 2)$]
]
]


=== Factor Differences of Squares

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

#math.equation(block: true, alt: "open parenthesis 3 x minus 4 close parenthesis open parenthesis 3 x plus 4 close parenthesis; 9 x squared minus 16")[$( 3 x − 4 ) ( 3 x + 4 ) \ 9 x^(2) − 16$]Remember, when you multiply conjugate binomials, the middle terms of the product add to 0. All you have left is a binomial, the difference of squares.

Multiplying conjugates is the only way to get a binomial from the product of two binomials.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product of Conjugates Pattern]

If #emph[a] and #emph[b] are real numbers

#math.equation(block: true, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared")[$( a − b ) ( a + b ) = a^(2) − b^(2)$]The product is called a difference of squares.
]

To factor, we will use the product pattern “in reverse” to factor the difference of squares. A #strong[difference of squares] factors to a product of conjugates.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Difference of Squares Pattern]

If #emph[a] and #emph[b] are real numbers,

#figure(figph[This image shows the difference of two squares formula, a squared – b squared = (a – b)(a + b). Also, the squares are labeled, a squared and b squared. The difference is shown between the two terms. Finally, the factoring (a – b)(a + b) are labeled as conjugates.], alt: "This image shows the difference of two squares formula, a squared – b squared = (a – b)(a + b). Also, the squares are labeled, a squared and b squared. The difference is shown between the two terms. Finally, the factoring (a – b)(a + b) are labeled as conjugates.", caption: none)
]

Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.

#examplebox("Example 6")[How to Factor Differences of Squares][
Factor: #math.equation(block: false, alt: "x squared minus 4")[$x^(2) − 4$].

#solutionbox[
#figure(figph[This table gives the steps for factoring x squared minus 4. The first step is identifying the pattern in the binomial including it is a difference. Also, the first and last terms are verified as squares.], alt: "This table gives the steps for factoring x squared minus 4. The first step is identifying the pattern in the binomial including it is a difference. Also, the first and last terms are verified as squares.", caption: none)

#figure(figph[The second step is writing the two terms as squares, x squared and 2 squared.], alt: "The second step is writing the two terms as squares, x squared and 2 squared.", caption: none)

#figure(figph[The second step is writing the two terms as squares, x squared and 2 squared. The third step is to write the factoring as a product of the conjugates (x – 2)(x + 2).], alt: "The second step is writing the two terms as squares, x squared and 2 squared. The third step is to write the factoring as a product of the conjugates (x – 2)(x + 2).", caption: none)

#figure(figph[The last step is to check with multiplication.], alt: "The last step is to check with multiplication.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "h squared minus 81")[$h^(2) − 81$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis h minus 9 close parenthesis open parenthesis h plus 9 close parenthesis")[$( h − 9 ) ( h + 9 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "k squared minus 121")[$k^(2) − 121$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis k minus 11 close parenthesis open parenthesis k plus 11 close parenthesis")[$( k − 11 ) ( k + 11 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor differences of squares.]

#math.equation(block: true, alt: "Step 1. Does the binomial fit the pattern?, a squared minus b squared; times Is this a difference?, ____ minus ____; times Are the first and last terms perfect squares?; Step 2. Write them as squares., open parenthesis a close parenthesis squared minus open parenthesis b close parenthesis squared; Step 3. Write the product of conjugates., open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis; Step 4. Check by multiplying.")[$"Step 1." #h(0.2em) "Does the binomial fit the pattern?" &  &  & a^(2) − b^(2) \ #h(2.5em) ⋅ #h(0.5em) "Is this a difference?" &  &  & \_\_\_\_ − \_\_\_\_ \ #h(2.5em) ⋅ #h(0.5em) "Are the first and last terms perfect squares?" &  &  &  \ "Step 2." #h(0.2em) "Write them as squares." &  &  & attach(( a ), t: 2) − attach(( b ), t: 2) \ "Step 3." #h(0.2em) "Write the product of conjugates." &  &  & ( a − b ) ( a + b ) \ "Step 4." #h(0.2em) "Check by multiplying." &  &  &$]
]

It is important to remember that #emph[sums of squares do not factor into a product of binomials]. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression #math.equation(block: false, alt: "a squared plus b squared")[$a^(2) + b^(2)$] is prime!

Don’t forget that 1 is a perfect square. We’ll need to use that fact in the next example.

#examplebox("Example 7")[][
Factor: #math.equation(block: false, alt: "64 y squared minus 1")[$64 y^(2) − 1$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the mathematical expression 64y^2 - 1, which represents a difference of squares and is a common form in algebra for factorization.], alt: "The image shows the mathematical expression 64y^2 - 1, which represents a difference of squares and is a common form in algebra for factorization.", caption: none)]),
  [Is this a difference? Yes.], [#figure(figph[The image displays the mathematical expression 64y^2 - 1, which represents a difference of squares problem in algebra.], alt: "The image displays the mathematical expression 64y^2 - 1, which represents a difference of squares problem in algebra.", caption: none)],
  [Are the first and last terms perfect squares?], [],
  [Yes - write them as squares.], [#figure(figph[Mathematical expressions demonstrating the difference of squares formula, with 'a^2 - b^2' in red and an example '(8y)^2 - 1^2' below it.], alt: "Mathematical expressions demonstrating the difference of squares formula, with 'a^2 - b^2' in red and an example '(8y)^2 - 1^2' below it.", caption: none)],
  [Factor as the product of conjugates.], [#figure(figph[Difference of squares formula (a-b)(a+b) exemplified by (8y-1)(8y+1).], alt: "Difference of squares formula (a-b)(a+b) exemplified by (8y-1)(8y+1).", caption: none)],
  [Check by multiplying.], [],
  [#math.equation(block: false, alt: "open parenthesis 8 y minus 1 close parenthesis open parenthesis 8 y plus 1 close parenthesis")[$( 8 y − 1 ) ( 8 y + 1 )$]], [],
  [#math.equation(block: false, alt: "64 y squared minus 1 ✓")[$64 y^(2) − 1 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "m squared minus 1")[$m^(2) − 1$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis m minus 1 close parenthesis open parenthesis m plus 1 close parenthesis")[$( m − 1 ) ( m + 1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "81 y squared minus 1")[$81 y^(2) − 1$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 9 y minus 1 close parenthesis open parenthesis 9 y plus 1 close parenthesis")[$( 9 y − 1 ) ( 9 y + 1 )$]
]
]

#examplebox("Example 8")[][
Factor: #math.equation(block: false, alt: "121 x squared minus 49 y squared")[$121 x^(2) − 49 y^(2)$].

#solutionbox[
#math.equation(block: true, alt: "121 x squared minus 49 y squared; Is this a difference of squares? Yes., open parenthesis 11 x close parenthesis squared minus open parenthesis 7 y close parenthesis squared; Factor as the product of conjugates., open parenthesis 11 x minus 7 y close parenthesis open parenthesis 11 x plus 7 y close parenthesis; Check by multiplying.; open parenthesis 11 x minus 7 y close parenthesis open parenthesis 11 x plus 7 y close parenthesis; 121 x squared minus 49 y squared ✓")[$&  &  & 121 x^(2) − 49 y^(2) \  \  \ "Is this a difference of squares? Yes." &  &  & attach(( 11 x ), t: 2) − attach(( 7 y ), t: 2) \  \  \ "Factor as the product of conjugates." &  &  & ( 11 x − 7 y ) ( 11 x + 7 y ) \  \  \ "Check by multiplying." &  &  &  \  \  \ #h(2.5em) ( 11 x − 7 y ) ( 11 x + 7 y ) &  &  &  \ #h(2.5em) 121 x^(2) − 49 y^(2) #h(0.2em) ✓ &  &  &$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "196 m squared minus 25 n squared")[$196 m^(2) − 25 n^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 14 m minus 5 n close parenthesis open parenthesis 14 m plus 5 n close parenthesis")[$( 14 m − 5 n ) ( 14 m + 5 n )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "144 p squared minus 9 q squared")[$144 p^(2) − 9 q^(2)$].

#solutionbox[
#math.equation(block: true, alt: "9 open parenthesis 4 p minus q close parenthesis open parenthesis 4 p plus q close parenthesis")[$9 ( 4 p − q ) ( 4 p + q )$]
]
]

The binomial in the next example may look “backwards,” but it’s still the difference of squares.

#examplebox("Example 9")[][
Factor: #math.equation(block: false, alt: "100 minus h squared")[$100 − h^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "100 minus h squared")[$100 − h^(2)$]],
  [Is this a difference of squares? Yes.], [#math.equation(block: false, alt: "open parenthesis 10 close parenthesis squared minus open parenthesis h close parenthesis squared")[$attach(( 10 ), t: 2) − attach(( h ), t: 2)$]],
  [Factor as the product of conjugates.], [#math.equation(block: false, alt: "open parenthesis 10 minus h close parenthesis open parenthesis 10 plus h close parenthesis")[$( 10 − h ) ( 10 + h )$]],
  [Check by multiplying. #linebreak() 
#math.equation(block: false, alt: "open parenthesis 10 minus h close parenthesis open parenthesis 10 plus h close parenthesis; 100 minus h squared ✓")[$#h(2.5em) ( 10 − h ) ( 10 + h ) \ #h(2.5em) 100 − h^(2) #h(0.2em) ✓$]], [],
  [Be careful not to rewrite the original expression as #math.equation(block: false, alt: "h squared minus 100")[$h^(2) − 100$].], [],
))

Factor #math.equation(block: false, alt: "h squared minus 100")[$h^(2) − 100$] on your own and then notice how the result differs from #math.equation(block: false, alt: "open parenthesis 10 minus h close parenthesis open parenthesis 10 plus h close parenthesis")[$( 10 − h ) ( 10 + h )$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "144 minus x squared")[$144 − x^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 12 minus x close parenthesis open parenthesis 12 plus x close parenthesis")[$( 12 − x ) ( 12 + x )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "169 minus p squared")[$169 − p^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 13 minus p close parenthesis open parenthesis 13 plus p close parenthesis")[$( 13 − p ) ( 13 + p )$]
]
]

To completely factor the binomial in the next example, we’ll factor a difference of squares twice!

#examplebox("Example 10")[][
Factor: #math.equation(block: false, alt: "x to the power 4 minus y to the power 4")[$x^(4) − y^(4)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x to the power 4 minus y to the power 4")[$x^(4) − y^(4)$]]),
  [Is this a difference of squares? Yes.], [#math.equation(block: false, alt: "open parenthesis x squared close parenthesis squared minus open parenthesis y squared close parenthesis squared")[$attach(( x^(2) ), t: 2) − attach(( y^(2) ), t: 2)$]],
  [Factor it as the product of conjugates.], [#math.equation(block: false, alt: "open parenthesis x squared minus y squared close parenthesis open parenthesis x squared plus y squared close parenthesis")[$( x^(2) − y^(2) ) ( x^(2) + y^(2) )$]],
  [Notice the first binomial is also a difference of squares!], [#math.equation(block: false, alt: "open parenthesis open parenthesis x close parenthesis squared minus open parenthesis y close parenthesis squared close parenthesis open parenthesis x squared plus y squared close parenthesis")[$( attach(( x ), t: 2) − attach(( y ), t: 2) ) ( x^(2) + y^(2) )$]],
  [Factor it as the product of conjugates. The last #linebreak() factor, the sum of squares, cannot be factored.], [#math.equation(block: false, alt: "open parenthesis x minus y close parenthesis open parenthesis x plus y close parenthesis open parenthesis x squared plus y squared close parenthesis")[$( x − y ) ( x + y ) ( x^(2) + y^(2) )$]],
  [Check by multiplying. #linebreak() 
#math.equation(block: false, alt: "open parenthesis x minus y close parenthesis open parenthesis x plus y close parenthesis open parenthesis x squared plus y squared close parenthesis; open bracket open parenthesis x minus y close parenthesis open parenthesis x plus y close parenthesis close bracket open parenthesis x squared plus y squared close parenthesis; open parenthesis x squared minus y squared close parenthesis open parenthesis x squared plus y squared close parenthesis; x to the power 4 minus y to the power 4 ✓")[$#h(2.5em) ( x − y ) ( x + y ) ( x^(2) + y^(2) ) \ #h(2.5em) [ ( x − y ) ( x + y ) ] ( x^(2) + y^(2) ) \ #h(2.5em) ( x^(2) − y^(2) ) ( x^(2) + y^(2) ) \ #h(2.5em) x^(4) − y^(4) #h(0.2em) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "a to the power 4 minus b to the power 4")[$a^(4) − b^(4)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a squared plus b squared close parenthesis open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis")[$( a^(2) + b^(2) ) ( a + b ) ( a − b )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "x to the power 4 minus 16")[$x^(4) − 16$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x squared plus 4 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x minus 2 close parenthesis")[$( x^(2) + 4 ) ( x + 2 ) ( x − 2 )$]
]
]

As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.

#examplebox("Example 11")[][
Factor: #math.equation(block: false, alt: "8 x squared y minus 98 y")[$8 x^(2) y − 98 y$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "8 x squared y minus 98 y")[$8 x^(2) y − 98 y$]]),
  [Is there a GCF? Yes, 2#emph[y]—factor it out!], [#math.equation(block: false, alt: "2 y open parenthesis 4 x squared minus 49 close parenthesis")[$2 y ( 4 x^(2) − 49 )$]],
  [Is the binomial a difference of squares? Yes.], [#math.equation(block: false, alt: "2 y open parenthesis open parenthesis 2 x close parenthesis squared minus open parenthesis 7 close parenthesis squared close parenthesis")[$2 y ( attach(( 2 x ), t: 2) − attach(( 7 ), t: 2) )$]],
  [Factor as a product of conjugates.], [#math.equation(block: false, alt: "2 y open parenthesis 2 x minus 7 close parenthesis open parenthesis 2 x plus 7 close parenthesis")[$2 y ( 2 x − 7 ) ( 2 x + 7 )$]],
  [Check by multiplying. #linebreak() #math.equation(block: false, alt: "2 y open parenthesis 2 x minus 7 close parenthesis open parenthesis 2 x plus 7 close parenthesis; 2 y open bracket open parenthesis 2 x minus 7 close parenthesis open parenthesis 2 x plus 7 close parenthesis close bracket; 2 y open parenthesis 4 x squared minus 49 close parenthesis; 8 x squared y minus 98 y ✓")[$#h(2.5em) 2 y ( 2 x − 7 ) ( 2 x + 7 ) \ #h(2.5em) 2 y [ ( 2 x − 7 ) ( 2 x + 7 ) ] \ #h(2.5em) 2 y ( 4 x^(2) − 49 ) \ #h(2.5em) 8 x^(2) y − 98 y #h(0.2em) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "7 x y squared minus 175 x")[$7 x y^(2) − 175 x$].

#solutionbox[
#math.equation(block: true, alt: "7 x open parenthesis y minus 5 close parenthesis open parenthesis y plus 5 close parenthesis")[$7 x ( y − 5 ) ( y + 5 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "45 a squared b minus 80 b")[$45 a^(2) b − 80 b$].

#solutionbox[
#math.equation(block: true, alt: "5 b open parenthesis 3 a minus 4 close parenthesis open parenthesis 3 a plus 4 close parenthesis")[$5 b ( 3 a − 4 ) ( 3 a + 4 )$]
]
]

#examplebox("Example 12")[][
Factor: #math.equation(block: false, alt: "6 x squared plus 96")[$6 x^(2) + 96$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "6 x squared plus 96")[$6 x^(2) + 96$]]),
  [Is there a GCF? Yes, 6—factor it out!], [#math.equation(block: false, alt: "6 open parenthesis x squared plus 16 close parenthesis")[$6 ( x^(2) + 16 )$]],
  [Is the binomial a difference of squares? No, it #linebreak() is a sum of squares. Sums of squares do not factor!], [],
  [Check by multiplying. #linebreak() #math.equation(block: false, alt: "6 open parenthesis x squared plus 16 close parenthesis; 6 x squared plus 96 ✓")[$#h(2.5em) 6 ( x^(2) + 16 ) \ #h(2.5em) 6 x^(2) + 96 #h(0.2em) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "8 a squared plus 200")[$8 a^(2) + 200$].

#solutionbox[
#math.equation(block: true, alt: "8 open parenthesis a squared plus 25 close parenthesis")[$8 ( a^(2) + 25 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "36 y squared plus 81")[$36 y^(2) + 81$].

#solutionbox[
#math.equation(block: true, alt: "9 open parenthesis 4 y squared plus 9 close parenthesis")[$9 ( 4 y^(2) + 9 )$]
]
]


=== Factor Sums and Differences of Cubes

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.

#math.equation(block: true, alt: "a cubed plus b cubed equals open parenthesis a plus b close parenthesis open parenthesis a squared minus a b plus b squared close parenthesis; a cubed minus b cubed equals open parenthesis a minus b close parenthesis open parenthesis a squared plus a b plus b squared close parenthesis")[$a^(3) + b^(3) = ( a + b ) ( a^(2) − a b + b^(2) ) \ a^(3) − b^(3) = ( a − b ) ( a^(2) + a b + b^(2) )$]We’ll check the first pattern and leave the second to you.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the algebraic expression (a + b)(a^2 - ab + b^2), which is a factored form of the sum of two cubes, a^3 + b^3. The first term (a+b) is highlighted in red.], alt: "The image shows the algebraic expression (a + b)(a^2 - ab + b^2), which is a factored form of the sum of two cubes, a^3 + b^3. The first term (a+b) is highlighted in red.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression reads a(a^2 - ab + b^2) + b(a^2 - ab + b^2), demonstrating the distributive property with 'a' and 'b' multiplying a common trinomial.], alt: "A mathematical expression reads a(a^2 - ab + b^2) + b(a^2 - ab + b^2), demonstrating the distributive property with 'a' and 'b' multiplying a common trinomial.", caption: none)],
  [Multiply.], [#math.equation(block: false, alt: "a cubed minus a squared b plus a b squared plus a squared b minus a b squared plus b cubed")[$a^(3) − a^(2) b + attach(a b, t: 2) + a^(2) b − attach(a b, t: 2) + b^(3)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "a cubed plus b cubed")[$a^(3) + b^(3)$]],
))

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Sum and Difference of Cubes Pattern]

#math.equation(block: true, alt: "a cubed plus b cubed equals open parenthesis a plus b close parenthesis open parenthesis a squared minus a b plus b squared close parenthesis; a cubed minus b cubed equals open parenthesis a minus b close parenthesis open parenthesis a squared plus a b plus b squared close parenthesis")[$a^(3) + b^(3) = ( a + b ) ( a^(2) − a b + b^(2) ) \ a^(3) − b^(3) = ( a − b ) ( a^(2) + a b + b^(2) )$]
]

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.

#figure(figph[This figure demonstrates the sign patterns in the sum and difference of two cubes. For the sum of two cubes, this figure shows the first two signs are plus and the first and the third signs are opposite, plus minus. The difference of two cubes has the first two signs the same, minus. The first and the third sign are minus plus.], alt: "This figure demonstrates the sign patterns in the sum and difference of two cubes. For the sum of two cubes, this figure shows the first two signs are plus and the first and the third signs are opposite, plus minus. The difference of two cubes has the first two signs the same, minus. The first and the third sign are minus plus.", caption: none)

The trinomial factor in the #strong[sum and difference of cubes pattern] cannot be factored.

It can be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in .

#figure(table(
  columns: 11, align: left, inset: 6pt,
  table.header([#emph[n]], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]),
  [#math.equation(block: false, alt: "n cubed")[$n^(3)$]], [1], [8], [27], [64], [125], [216], [343], [512], [729], [1000],
))

#examplebox("Example 13")[How to Factor the Sum or Difference of Cubes][
Factor: #math.equation(block: false, alt: "x cubed plus 64")[$x^(3) + 64$].

#solutionbox[
#figure(figph[This table gives the steps for factoring x cubed + 64. The first step is to verify the binomial fits the pattern. Also, to check the sign for a sum or difference. This binomial is a sum that fits the pattern.], alt: "This table gives the steps for factoring x cubed + 64. The first step is to verify the binomial fits the pattern. Also, to check the sign for a sum or difference. This binomial is a sum that fits the pattern.", caption: none)

#figure(figph[The second step is to write the terms as cubes, x cubed + 4 cubed.], alt: "The second step is to write the terms as cubes, x cubed + 4 cubed.", caption: none)

#figure(figph[The third step is follow the pattern for the sum of two cubes, (x + 4)(x squared minus x times 4 + 4 squared).], alt: "The third step is follow the pattern for the sum of two cubes, (x + 4)(x squared minus x times 4 + 4 squared).", caption: none)

#figure(figph[The fourth step is to simplify, (x + 4)(x squared minus 4 x +16).], alt: "The fourth step is to simplify, (x + 4)(x squared minus 4 x +16).", caption: none)

#figure(figph[The last step is to check the answer with multiplication.], alt: "The last step is to check the answer with multiplication.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "x cubed plus 27")[$x^(3) + 27$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x squared minus 3 x plus 9 close parenthesis")[$( x + 3 ) ( x^(2) − 3 x + 9 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "y cubed plus 8")[$y^(3) + 8$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis y plus 2 close parenthesis open parenthesis y squared minus 2 y plus 4 close parenthesis")[$( y + 2 ) ( y^(2) − 2 y + 4 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor the sum or difference of cubes.]

To factor the sum or difference of cubes:

+ Does the binomial fit the sum or difference of cubes pattern?
- Is it a sum or difference?
- Are the first and last terms perfect cubes?
+ Write them as cubes.
+ Use either the sum or difference of cubes pattern.
+ Simplify inside the parentheses
+ Check by multiplying the factors.
]

#examplebox("Example 14")[][
Factor: #math.equation(block: false, alt: "x cubed minus 1000")[$x^(3) − 1000$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression x cubed minus 1000 is displayed on a white background.], alt: "The mathematical expression x cubed minus 1000 is displayed on a white background.", caption: none)]),
  [This binomial is a difference. The first and last terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[Two mathematical expressions are shown: a^3 - b^3 in red, and x^3 - 10^3 in black below it, illustrating the difference of cubes.], alt: "Two mathematical expressions are shown: a^3 - b^3 in red, and x^3 - 10^3 in black below it, illustrating the difference of cubes.", caption: none)],
  [Use the difference of cubes pattern.], [#figure(figph[(a - b)(a^2 + ab + b^2) and (x - 10)(x^2 + 10 \* x + 10^2) are shown, illustrating the difference of cubes factorization.], alt: "(a - b)(a^2 + ab + b^2) and (x - 10)(x^2 + 10 * x + 10^2) are shown, illustrating the difference of cubes factorization.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the difference of cubes factorization: (a-b)(a^2+ab+b^2) in red, with a specific example (x-10)(x^2+10x+100) in black below it.], alt: "The image shows the difference of cubes factorization: (a-b)(a^2+ab+b^2) in red, with a specific example (x-10)(x^2+10x+100) in black below it.", caption: none)],
  [Check by multiplying.], [],
  [#figure(figph[Polynomial long multiplication demonstrating the difference of cubes identity, (a-b)(a^2+ab+b^2) = a^3-b^3. The example shows (x-10)(x^2+10x+100) simplifying to x^3-1000.], alt: "Polynomial long multiplication demonstrating the difference of cubes identity, (a-b)(a^2+ab+b^2) = a^3-b^3. The example shows (x-10)(x^2+10x+100) simplifying to x^3-1000.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "u cubed minus 125")[$u^(3) − 125$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis u minus 5 close parenthesis open parenthesis u squared plus 5 u plus 25 close parenthesis")[$( u − 5 ) ( u^(2) + 5 u + 25 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "v cubed minus 343")[$v^(3) − 343$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis v minus 7 close parenthesis open parenthesis v squared plus 7 v plus 49 close parenthesis")[$( v − 7 ) ( v^(2) + 7 v + 49 )$]
]
]

Be careful to use the correct signs in the factors of the sum and difference of cubes.

#examplebox("Example 15")[][
Factor: #math.equation(block: false, alt: "512 minus 125 p cubed")[$512 − 125 p^(3)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the algebraic expression 512 - 125p^3.], alt: "The image shows the algebraic expression 512 - 125p^3.", caption: none)]),
  [This binomial is a difference. The first and last terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[A mathematical expression displaying the difference of cubes formula a³ - b³ above its application to 8³ - (5p)³.], alt: "A mathematical expression displaying the difference of cubes formula a³ - b³ above its application to 8³ - (5p)³.", caption: none)],
  [Use the difference of cubes pattern.], [#figure(figph[The image illustrates the difference of cubes factorization formula, (a - b)(a^2 + ab + b^2), with a specific example: (8 - 5p)(8^2 + 8 \* 5p + (5p)^2).], alt: "The image illustrates the difference of cubes factorization formula, (a - b)(a^2 + ab + b^2), with a specific example: (8 - 5p)(8^2 + 8 * 5p + (5p)^2).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the difference of cubes factorization: (8 - 5p)(64 + 40p + 25p^2) with the formula (a - b)(a^2 + ab + b^2) highlighted.], alt: "A mathematical expression showing the difference of cubes factorization: (8 - 5p)(64 + 40p + 25p^2) with the formula (a - b)(a^2 + ab + b^2) highlighted.", caption: none)],
  [Check by multiplying.], [We'll leave the check to you.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "64 minus 27 x cubed")[$64 − 27 x^(3)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 minus 3 x close parenthesis open parenthesis 16 plus 12 x plus 9 x squared close parenthesis")[$( 4 − 3 x ) ( 16 + 12 x + 9 x^(2) )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "27 minus 8 y cubed")[$27 − 8 y^(3)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 minus 2 y close parenthesis open parenthesis 9 plus 6 y plus 4 y squared close parenthesis")[$( 3 − 2 y ) ( 9 + 6 y + 4 y^(2) )$]
]
]

#examplebox("Example 16")[][
Factor: #math.equation(block: false, alt: "27 u cubed minus 125 v cubed")[$27 u^(3) − 125 v^(3)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the difference of two cubes: 27u^3 - 125v^3. This is a common form in algebra for factorization problems.], alt: "A mathematical expression showing the difference of two cubes: 27u^3 - 125v^3. This is a common form in algebra for factorization problems.", caption: none)]),
  [This binomial is a difference. The first and last terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[A mathematical expression showing the difference of cubes formula a^3 - b^3, followed by an example with specific values: (3u)^3 - (5v)^3.], alt: "A mathematical expression showing the difference of cubes formula a^3 - b^3, followed by an example with specific values: (3u)^3 - (5v)^3.", caption: none)],
  [Use the difference of cubes pattern.], [#figure(figph[An algebraic expression demonstrating the difference of cubes formula: (a - b)(a^2 + ab + b^2) = a^3 - b^3, with 'a' represented by 3u and 'b' by 5v.], alt: "An algebraic expression demonstrating the difference of cubes formula: (a - b)(a^2 + ab + b^2) = a^3 - b^3, with 'a' represented by 3u and 'b' by 5v.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the product of two binomials, (3u - 5v) and (9u^2 + 15uv + 25v^2), with the difference of cubes formula (a - b)(a^2 + ab + b^2) highlighted in red.], alt: "A mathematical expression showing the product of two binomials, (3u - 5v) and (9u^2 + 15uv + 25v^2), with the difference of cubes formula (a - b)(a^2 + ab + b^2) highlighted in red.", caption: none)],
  [Check by multiplying.], [We'll leave the check to you.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "8 x cubed minus 27 y cubed")[$8 x^(3) − 27 y^(3)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 x minus 3 y close parenthesis open parenthesis 4 x squared plus 6 x y plus 9 y squared close parenthesis")[$( 2 x − 3 y ) ( 4 x^(2) + 6 x y + 9 y^(2) )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "1000 m cubed minus 125 n cubed")[$1000 m^(3) − 125 n^(3)$].

#solutionbox[
#math.equation(block: true, alt: "125 open parenthesis 4 m squared plus 2 m n plus n squared close parenthesis open parenthesis 2 m minus n close parenthesis")[$125 ( 4 m^(2) + 2 m n + n^(2) ) ( 2 m − n )$]
]
]

In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.

#examplebox("Example 17")[][
Factor: #math.equation(block: false, alt: "5 m cubed plus 40 n cubed")[$5 m^(3) + 40 n^(3)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays 5m^3 + 40n^3, written in a clear, digital font on a white background, suggesting an algebraic problem or formula.], alt: "A mathematical expression displays 5m^3 + 40n^3, written in a clear, digital font on a white background, suggesting an algebraic problem or formula.", caption: none)]),
  [Factor the common factor.], [#figure(figph[The image displays the algebraic expression 5(m^3 + 8n^3) on a white background. The expression is written in a clear, standard mathematical font.], alt: "The image displays the algebraic expression 5(m^3 + 8n^3) on a white background. The expression is written in a clear, standard mathematical font.", caption: none)],
  [This binomial is a sum. The first and last terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[A mathematical expression showing 5 times the sum of m cubed and the quantity 2n cubed, with red 'a cubed' and 'b cubed' labels above m cubed and (2n) cubed respectively.], alt: "A mathematical expression showing 5 times the sum of m cubed and the quantity 2n cubed, with red 'a cubed' and 'b cubed' labels above m cubed and (2n) cubed respectively.", caption: none)],
  [Use the sum of cubes pattern.], [#figure(figph[Mathematical expression 5(m+2n)(m²-m\*2n+(2n)²), illustrating the sum of cubes factorization. Variables 'a' and 'b' (m and 2n respectively) are highlighted in red.], alt: "Mathematical expression 5(m+2n)(m²-m*2n+(2n)²), illustrating the sum of cubes factorization. Variables 'a' and 'b' (m and 2n respectively) are highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[An algebraic expression showing 5 multiplied by the product of (m + 2n) and (m^2 - 2mn + 4n^2), illustrating the factorization pattern for the sum of two cubes, a^3 + b^3.], alt: "An algebraic expression showing 5 multiplied by the product of (m + 2n) and (m^2 - 2mn + 4n^2), illustrating the factorization pattern for the sum of two cubes, a^3 + b^3.", caption: none)],
))

Check. To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by 5. We’ll leave the multiplication for you.

#math.equation(block: true, alt: "5 open parenthesis m plus 2 n close parenthesis open parenthesis m squared minus 2 m n plus 4 n squared close parenthesis")[$5 ( limits(m + 2 n)^ ) ( limits(m^(2) − 2 m n + 4 n^(2))^ )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "500 p cubed plus 4 q cubed")[$500 p^(3) + 4 q^(3)$].

#solutionbox[
#math.equation(block: true, alt: "4 open parenthesis 5 p plus q close parenthesis open parenthesis 25 p squared minus 5 p q plus q squared close parenthesis")[$4 ( 5 p + q ) ( 25 p^(2) − 5 p q + q^(2) )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "432 c cubed plus 686 d cubed")[$432 c^(3) + 686 d^(3)$].

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis 6 c plus 7 d close parenthesis open parenthesis 36 c squared minus 42 c d plus 49 d squared close parenthesis")[$2 ( 6 c + 7 d ) ( 36 c^(2) − 42 c d + 49 d^(2) )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with factoring special products.

- #link("https://openstax.org/l/25SumCubes")[Sum or Difference of Cubes]
- #link("https://openstax.org/l/25DiffCubes")[Difference of Cubes Factoring]
]


=== Key Concepts

- #strong[Factor perfect square trinomials] #linebreak() 
#math.equation(block: false, alt: "Step 1. Does the trinomial fit the pattern?, a squared plus 2 a b plus b squared, a squared minus 2 a b plus b squared; Is the first term a perfect square?, open parenthesis a close parenthesis squared, open parenthesis a close parenthesis squared; Write it as a square.; Is the last term a perfect square?, open parenthesis a close parenthesis squared open parenthesis b close parenthesis squared, open parenthesis a close parenthesis squared open parenthesis b close parenthesis squared; Write it as a square.; Check the middle term. Is it 2 a b ?, open parenthesis a close parenthesis squared ↘ 2 times a times b ↙ open parenthesis b close parenthesis squared, open parenthesis a close parenthesis squared ↘ 2 times a times b ↙ open parenthesis b close parenthesis squared; Step 2. Write the squar")[$"Step 1." #h(0.2em) "Does the trinomial fit the pattern?" &  &  & a^(2) + 2 a b + b^(2) &  &  & a^(2) − 2 a b + b^(2) \ #h(3em) "Is the first term a perfect square?" &  &  & attach(( a ), t: 2) &  &  & attach(( a ), t: 2) \ #h(3em) "Write it as a square." &  &  &  &  &  &  \ #h(3em) "Is the last term a perfect square?" &  &  & attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) &  &  & attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) \ #h(3em) "Write it as a square." &  &  &  &  &  &  \ #h(3em) "Check the middle term. Is it" #h(0.2em) 2 a b ? &  &  & attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) &  &  & attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) \ "Step 2." #h(0.2em) "Write the square of the binomial." &  &  & attach(( a + b ), t: 2) &  &  & attach(( a − b ), t: 2) \ "Step 3." #h(0.2em) "Check by multiplying." &  &  &  &  &  &$]
- #strong[Factor differences of squares] #linebreak() 
#math.equation(block: false, alt: "Step 1. Does the binomial fit the pattern?, a squared minus b squared; Is this a difference?, ____ minus ____; Are the first and last terms perfect squares?; Step 2. Write them as squares., open parenthesis a close parenthesis squared minus open parenthesis b close parenthesis squared; Step 3. Write the product of conjugates., open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis; Step 4. Check by multiplying.")[$"Step 1." #h(0.2em) "Does the binomial fit the pattern?" &  &  & a^(2) − b^(2) \ #h(3em) "Is this a difference?" &  &  & \_\_\_\_ − \_\_\_\_ \ #h(3em) "Are the first and last terms perfect squares?" &  &  &  \ "Step 2." #h(0.2em) "Write them as squares." &  &  & attach(( a ), t: 2) − attach(( b ), t: 2) \ "Step 3." #h(0.2em) "Write the product of conjugates." &  &  & ( a − b ) ( a + b ) \ "Step 4." #h(0.2em) "Check by multiplying." &  &  &$]
- #strong[Factor sum and difference of cubes] To factor the sum or difference of cubes:
+ Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes?
+ Write them as cubes.
+ Use either the sum or difference of cubes pattern.
+ Simplify inside the parentheses
+ Check by multiplying the factors.


==== Practice Makes Perfect

#strong[Factor Perfect Square Trinomials]

In the following exercises, factor.

#math.equation(block: true, alt: "16 y squared plus 24 y plus 9")[$16 y^(2) + 24 y + 9$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 y plus 3 close parenthesis squared")[$attach(( 4 y + 3 ), t: 2)$]
]

#math.equation(block: true, alt: "25 v squared plus 20 v plus 4")[$25 v^(2) + 20 v + 4$]

#math.equation(block: true, alt: "36 s squared plus 84 s plus 49")[$36 s^(2) + 84 s + 49$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 6 s plus 7 close parenthesis squared")[$attach(( 6 s + 7 ), t: 2)$]
]

#math.equation(block: true, alt: "49 s squared plus 154 s plus 121")[$49 s^(2) + 154 s + 121$]

#math.equation(block: true, alt: "100 x squared minus 20 x plus 1")[$100 x^(2) − 20 x + 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 10 x minus 1 close parenthesis squared")[$attach(( 10 x − 1 ), t: 2)$]
]

#math.equation(block: true, alt: "64 z squared minus 16 z plus 1")[$64 z^(2) − 16 z + 1$]

#math.equation(block: true, alt: "25 n squared minus 120 n plus 144")[$25 n^(2) − 120 n + 144$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 n minus 12 close parenthesis squared")[$attach(( 5 n − 12 ), t: 2)$]
]

#math.equation(block: true, alt: "4 p squared minus 52 p plus 169")[$4 p^(2) − 52 p + 169$]

#math.equation(block: true, alt: "49 x squared minus 28 x y plus 4 y squared")[$49 x^(2) − 28 x y + 4 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 7 x minus 2 y close parenthesis squared")[$attach(( 7 x − 2 y ), t: 2)$]
]

#math.equation(block: true, alt: "25 r squared minus 60 r s plus 36 s squared")[$25 r^(2) − 60 r s + 36 s^(2)$]

#math.equation(block: true, alt: "25 n squared plus 25 n plus 4")[$25 n^(2) + 25 n + 4$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 n plus 4 close parenthesis open parenthesis 5 n plus 1 close parenthesis")[$( 5 n + 4 ) ( 5 n + 1 )$]
]

#math.equation(block: true, alt: "100 y squared minus 20 y plus 1")[$100 y^(2) − 20 y + 1$]

#math.equation(block: true, alt: "64 m squared minus 16 m plus 1")[$64 m^(2) − 16 m + 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 8 m minus 1 close parenthesis squared")[$( 8 m − 1 )^(2)$]
]

#math.equation(block: true, alt: "100 x squared minus 25 x plus 1")[$100 x^(2) − 25 x + 1$]

#math.equation(block: true, alt: "10 k squared plus 80 k plus 160")[$10 k^(2) + 80 k + 160$]

#solutionbox[
#math.equation(block: true, alt: "10 open parenthesis k plus 4 close parenthesis squared")[$10 attach(( k + 4 ), t: 2)$]
]

#math.equation(block: true, alt: "64 x squared minus 96 x plus 36")[$64 x^(2) − 96 x + 36$]

#math.equation(block: true, alt: "75 u cubed minus 30 u squared v plus 3 u v squared")[$75 u^(3) − 30 u^(2) v + 3 u v^(2)$]

#solutionbox[
#math.equation(block: true, alt: "3 u open parenthesis 5 u minus v close parenthesis squared")[$3 u attach(( 5 u − v ), t: 2)$]
]

#math.equation(block: true, alt: "90 p cubed plus 300 p squared q plus 250 p q squared")[$90 p^(3) + 300 p^(2) q + 250 p q^(2)$]

#strong[Factor Differences of Squares]

In the following exercises, factor.

#math.equation(block: true, alt: "x squared minus 16")[$x^(2) − 16$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis")[$( x − 4 ) ( x + 4 )$]
]

#math.equation(block: true, alt: "n squared minus 9")[$n^(2) − 9$]

#math.equation(block: true, alt: "25 v squared minus 1")[$25 v^(2) − 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 v minus 1 close parenthesis open parenthesis 5 v plus 1 close parenthesis")[$( 5 v − 1 ) ( 5 v + 1 )$]
]

#math.equation(block: true, alt: "169 q squared minus 1")[$169 q^(2) − 1$]

#math.equation(block: true, alt: "121 x squared minus 144 y squared")[$121 x^(2) − 144 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 11 x minus 12 y close parenthesis open parenthesis 11 x plus 12 y close parenthesis")[$( 11 x − 12 y ) ( 11 x + 12 y )$]
]

#math.equation(block: true, alt: "49 x squared minus 81 y squared")[$49 x^(2) − 81 y^(2)$]

#math.equation(block: true, alt: "169 c squared minus 36 d squared")[$169 c^(2) − 36 d^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 13 c minus 6 d close parenthesis open parenthesis 13 c plus 6 d close parenthesis")[$( 13 c − 6 d ) ( 13 c + 6 d )$]
]

#math.equation(block: true, alt: "36 p squared minus 49 q squared")[$36 p^(2) − 49 q^(2)$]

#math.equation(block: true, alt: "4 minus 49 x squared")[$4 − 49 x^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 minus 7 x close parenthesis open parenthesis 2 plus 7 x close parenthesis")[$( 2 − 7 x ) ( 2 + 7 x )$]
]

#math.equation(block: true, alt: "121 minus 25 s squared")[$121 − 25 s^(2)$]

#math.equation(block: true, alt: "16 z to the power 4 minus 1")[$16 z^(4) − 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 z minus 1 close parenthesis open parenthesis 2 z plus 1 close parenthesis open parenthesis 4 z squared plus 1 close parenthesis")[$( 2 z − 1 ) ( 2 z + 1 ) ( 4 z^(2) + 1 )$]
]

#math.equation(block: true, alt: "m to the power 4 minus n to the power 4")[$m^(4) − n^(4)$]

#math.equation(block: true, alt: "5 q squared minus 45")[$5 q^(2) − 45$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis q minus 3 close parenthesis open parenthesis q plus 3 close parenthesis")[$5 ( q − 3 ) ( q + 3 )$]
]

#math.equation(block: true, alt: "98 r cubed minus 72 r")[$98 r^(3) − 72 r$]

#math.equation(block: true, alt: "24 p squared plus 54")[$24 p^(2) + 54$]

#solutionbox[
#math.equation(block: true, alt: "6 open parenthesis 4 p squared plus 9 close parenthesis")[$6 ( 4 p^(2) + 9 )$]
]

#math.equation(block: true, alt: "20 b squared plus 140")[$20 b^(2) + 140$]

#strong[Factor Sums and Differences of Cubes]

In the following exercises, factor.

#math.equation(block: true, alt: "x cubed plus 125")[$x^(3) + 125$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 5 close parenthesis open parenthesis x squared minus 5 x plus 25 close parenthesis")[$( x + 5 ) ( x^(2) − 5 x + 25 )$]
]

#math.equation(block: true, alt: "n cubed plus 512")[$n^(3) + 512$]

#math.equation(block: true, alt: "z cubed minus 27")[$z^(3) − 27$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis z minus 3 close parenthesis open parenthesis z squared plus 3 z plus 9 close parenthesis")[$( z − 3 ) ( z^(2) + 3 z + 9 )$]
]

#math.equation(block: true, alt: "v cubed minus 216")[$v^(3) − 216$]

#math.equation(block: true, alt: "8 minus 343 t cubed")[$8 − 343 t^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 minus 7 t close parenthesis open parenthesis 4 plus 14 t plus 49 t squared close parenthesis")[$( 2 − 7 t ) ( 4 + 14 t + 49 t^(2) )$]
]

#math.equation(block: true, alt: "125 minus 27 w cubed")[$125 − 27 w^(3)$]

#math.equation(block: true, alt: "8 y cubed minus 125 z cubed")[$8 y^(3) − 125 z^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 y minus 5 z close parenthesis open parenthesis 4 y squared plus 10 y z plus 25 z squared close parenthesis")[$( 2 y − 5 z ) ( 4 y^(2) + 10 y z + 25 z^(2) )$]
]

#math.equation(block: true, alt: "27 x cubed minus 64 y cubed")[$27 x^(3) − 64 y^(3)$]

#math.equation(block: true, alt: "7 k cubed plus 56")[$7 k^(3) + 56$]

#solutionbox[
#math.equation(block: true, alt: "7 open parenthesis k plus 2 close parenthesis open parenthesis k squared minus 2 k plus 4 close parenthesis")[$7 ( k + 2 ) ( k^(2) − 2 k + 4 )$]
]

#math.equation(block: true, alt: "6 x cubed minus 48 y cubed")[$6 x^(3) − 48 y^(3)$]

#math.equation(block: true, alt: "2 minus 16 y cubed")[$2 − 16 y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis 1 minus 2 y close parenthesis open parenthesis 1 plus 2 y plus 4 y squared close parenthesis")[$2 ( 1 − 2 y ) ( 1 + 2 y + 4 y^(2) )$]
]

#math.equation(block: true, alt: "−2 x cubed minus 16 y cubed")[$−2 x^(3) − 16 y^(3)$]

#strong[Mixed Practice]

In the following exercises, factor.

#math.equation(block: true, alt: "64 a squared minus 25")[$64 a^(2) − 25$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 8 a minus 5 close parenthesis open parenthesis 8 a plus 5 close parenthesis")[$( 8 a − 5 ) ( 8 a + 5 )$]
]

#math.equation(block: true, alt: "121 x squared minus 144")[$121 x^(2) − 144$]

#math.equation(block: true, alt: "27 q squared minus 3")[$27 q^(2) − 3$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 3 q minus 1 close parenthesis open parenthesis 3 q plus 1 close parenthesis")[$3 ( 3 q − 1 ) ( 3 q + 1 )$]
]

#math.equation(block: true, alt: "4 p squared minus 100")[$4 p^(2) − 100$]

#math.equation(block: true, alt: "16 x squared minus 72 x plus 81")[$16 x^(2) − 72 x + 81$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 x minus 9 close parenthesis squared")[$attach(( 4 x − 9 ), t: 2)$]
]

#math.equation(block: true, alt: "36 y squared plus 12 y plus 1")[$36 y^(2) + 12 y + 1$]

#math.equation(block: true, alt: "8 p squared plus 2")[$8 p^(2) + 2$]

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis 4 p squared plus 1 close parenthesis")[$2 ( 4 p^(2) + 1 )$]
]

#math.equation(block: true, alt: "81 x squared plus 169")[$81 x^(2) + 169$]

#math.equation(block: true, alt: "125 minus 8 y cubed")[$125 − 8 y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 minus 2 y close parenthesis open parenthesis 25 plus 10 y plus 4 y squared close parenthesis")[$( 5 − 2 y ) ( 25 + 10 y + 4 y^(2) )$]
]

#math.equation(block: true, alt: "27 u cubed plus 1000")[$27 u^(3) + 1000$]

#math.equation(block: true, alt: "45 n squared plus 60 n plus 20")[$45 n^(2) + 60 n + 20$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis 3 n plus 2 close parenthesis squared")[$5 attach(( 3 n + 2 ), t: 2)$]
]

#math.equation(block: true, alt: "48 q cubed minus 24 q squared plus 3 q")[$48 q^(3) − 24 q^(2) + 3 q$]


==== Everyday Math

#strong[Landscaping] Sue and Alan are planning to put a 15 foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is #emph[w], the total area of the pool and deck is given by the trinomial #math.equation(block: false, alt: "4 w squared plus 60 w plus 225")[$4 w^(2) + 60 w + 225$]. Factor the trinomial.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 w plus 15 close parenthesis squared")[$attach(( 2 w + 15 ), t: 2)$]
]

#strong[Home repair] The height a twelve foot ladder can reach up the side of a building if the ladder’s base is #emph[b] feet from the building is the square root of the binomial #math.equation(block: false, alt: "144 minus b squared")[$144 − b^(2)$]. Factor the binomial.


==== Writing Exercises

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

#solutionbox[
Answers may vary.
]

How do you recognize the binomial squares pattern?

Explain why #math.equation(block: false, alt: "n squared plus 25 not equal to open parenthesis n plus 5 close parenthesis squared")[$n^(2) + 25 ≠ attach(( n + 5 ), t: 2)$]. Use algebra, words, or pictures.

#solutionbox[
Answers may vary.
]

Maribel factored #math.equation(block: false, alt: "y squared minus 30 y plus 81")[$y^(2) − 30 y + 81$] as #math.equation(block: false, alt: "open parenthesis y minus 9 close parenthesis squared")[$attach(( y − 9 ), t: 2)$]. Was she right or wrong? How do you know?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has the following statements all to be preceded by “I can…”. The first row is “factor perfect square trinomials”. The second row is “factor differences of squares”. The third row is “factor sums and differences of cubes”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.], alt: "This table has the following statements all to be preceded by “I can…”. The first row is “factor perfect square trinomials”. The second row is “factor differences of squares”. The third row is “factor sums and differences of cubes”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?