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== 6.3#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 3 x squared close parenthesis cubed .")[$attach(( 3 x^(2) ), t: 3) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "27 x to the power 6")[$27 x^(6)$]
]
]

#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 x minus 3 close parenthesis open parenthesis x plus 3 close parenthesis .")[$( x − 3 ) ( x + 3 ) .$] #linebreak() 
If you missed this problem, review .

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

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. We squared a binomial using the Binomial Squares pattern in a previous chapter.

#figure(figph[In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.], alt: "In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.", caption: none)

The trinomial #math.equation(block: false, alt: "9 x squared plus 24 x plus 16")[$9 x^(2) + 24 x + 16$] is called a #emph[perfect square trinomial]. It is the square of the binomial #math.equation(block: false, alt: "3 x plus 4.")[$3 x + 4 .$]

In this chapter, you will start with a perfect square trinomial and factor it into its #strong[prime] factors.

You could factor this #strong[trinomial] using the methods described in the last section, since it is of the form #math.equation(block: false, alt: "a x squared plus b x plus c .")[$a x^(2) + b x + c .$] But if you recognize that the first and last terms are squares and the trinomial fits the 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")[$a^(2) + 2 a b + b^(2) = attach(( a + b ), t: 2) \ a^(2) − 2 a b + b^(2) = attach(( a − b ), t: 2)$]
]

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 the product, #math.equation(block: false, alt: "2 a b ?")[$2 a b ?$] 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[Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.], alt: "Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.", caption: none)

#figure(figph[Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.], alt: "Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.", caption: none)

#figure(figph[Step 3 is to check by multiplying.], alt: "Step 3 is to check by multiplying.", 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; 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 squ")[$"Step 1." & "Does the trinomial fit the pattern?" &  &  & #h(4em) a^(2) + 2 a b + b^(2) &  &  & a^(2) − 2 a b + b^(2) \  & "Is the first term a perfect square?" &  &  & #h(4em) attach(( a ), t: 2) &  &  & attach(( a ), t: 2) \  & "Write it as a square." &  &  &  &  &  &  \  & "Is the last term a perfect square?" &  &  & #h(4em) attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) &  &  & attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) \  & "Write it as a square." &  &  &  &  &  &  \  & "Check the middle term. Is it" #h(0.2em) 2 a b ? &  &  & #h(4em) attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) &  &  & attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) \ "Step 2." & "Write the square of the binomial." &  &  & #h(4em) attach(( a + b ), t: 2) &  &  & attach(( a − b ), t: 2) \ "Step 3." & "Check by multiplying." &  &  &  &  &  &$]
]

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 #strong[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 shows the quadratic polynomial 81y^2 - 72y + 16, which is a perfect square trinomial.], alt: "A mathematical expression shows the quadratic polynomial 81y^2 - 72y + 16, which is a perfect square trinomial.", caption: none)]),
  [Are the first and last terms perfect squares?], [#figure(figph[Two mathematical expressions are shown: (9y)^2 on the left and (4)^2 on the right.], alt: "Two mathematical expressions are shown: (9y)^2 on the left and (4)^2 on the right.", caption: none)],
  [Check the middle term.], [#figure(figph[A mathematical image illustrates the expansion of a binomial squared. The expressions (9y)^2 and (4)^2 point to 2(9y)(4), which simplifies to 72y. This shows the calculation of the 2ab term.], alt: "A mathematical image illustrates the expansion of a binomial squared. The expressions (9y)^2 and (4)^2 point to 2(9y)(4), which simplifies to 72y. This shows the calculation of the 2ab term.", 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[An example of a perfect square trinomial (a-b)^2 = a^2 - 2ab + b^2, specifically (9y)^2 - 2 \* 9y \* 4 + 4^2.], alt: "An example of a perfect square trinomial (a-b)^2 = a^2 - 2ab + b^2, specifically (9y)^2 - 2 * 9y * 4 + 4^2.", caption: none)],
  [Write as the square of a binomial.], [#figure(figph[The mathematical expression (9y-4)^2 is shown in black text on a white background. It represents the quantity (9y minus 4) raised to the power of 2, indicating that the binomial should be squared.], alt: "The mathematical expression (9y-4)^2 is shown in black text on a white background. It represents the quantity (9y minus 4) raised to the power of 2, indicating that the binomial should be squared.", caption: none)],
  [Check by multiplying: #linebreak()  #linebreak() 
#math.equation(block: false, alt: "open parenthesis 9 y minus 4 close parenthesis squared; open parenthesis 9 y close parenthesis squared minus 2 times 9 y times 4 plus 4 squared; 81 y squared minus 72 y plus 16 ✓")[$#h(4em) attach(( 9 y − 4 ), t: 2) \ attach(( 9 y ), t: 2) − 2 · 9 y · 4 + 4^(2) \ 81 y^(2) − 72 y + 16 ✓$]], [],
))
]
]

#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[A mathematical expression displaying the quadratic trinomial 36x^2 + 84xy + 49y^2.], alt: "A mathematical expression displaying the quadratic trinomial 36x^2 + 84xy + 49y^2.", caption: none)]),
  [Test each term to verify the pattern.], [#figure(figph[The perfect square trinomial formula (a^2 + 2ab + b^2) demonstrated with an example: (6x)^2 + 2(6x)(7y) + (7y)^2.], alt: "The perfect square trinomial formula (a^2 + 2ab + b^2) demonstrated with an example: (6x)^2 + 2(6x)(7y) + (7y)^2.", caption: none)],
  [Factor.], [#figure(figph[The mathematical expression (6x + 7y)    ² is centered on a white background, representing the square of a binomial.], alt: "The mathematical expression (6x + 7y) ² is centered on a white background, representing the square of a binomial.", caption: none)],
  [Check by multiplying. #linebreak()  #linebreak() #math.equation(block: false, alt: "open parenthesis 6 x plus 7 y close parenthesis squared; open parenthesis 6 x close parenthesis squared plus 2 times 6 x times 7 y plus open parenthesis 7 y close parenthesis squared; 36 x squared plus 84 x y plus 49 y squared ✓")[$#h(4em) attach(( 6 x + 7 y ), t: 2) \ attach(( 6 x ), t: 2) + 2 · 6 x · 7 y + attach(( 7 y ), t: 2) \ 36 x^(2) + 84 x y + 49 y^(2) ✓$]], [],
))
]
]

#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")[$attach(( 8 m + 7 n ), t: 2)$]
]
]

Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a #strong[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 4")[][
Factor: #math.equation(block: false, alt: "100 x squared y minus 80 x y plus 16 y .")[$100 x^(2) y − 80 x y + 16 y .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the algebraic expression 100x^2y - 80xy + 16y. This is a trinomial with three terms, each containing variables x and y, and constant coefficients.], alt: "The image displays the algebraic expression 100x^2y - 80xy + 16y. This is a trinomial with three terms, each containing variables x and y, and constant coefficients.", caption: none)]),
  [Is there a GCF? Yes, #math.equation(block: false, alt: "4 y ,")[$4 y ,$] so factor it out.], [#figure(figph[The mathematical expression 4y(25x^2 - 20x + 4) is displayed.], alt: "The mathematical expression 4y(25x^2 - 20x + 4) is displayed.", caption: none)],
  [Is this a perfect square trinomial?], [],
  [Verify the pattern.], [#figure(figph[A mathematical expression showing the expansion of a perfect square trinomial inside brackets, specifically 4y\[(5x)^2 - 2 \* 5x \* 2 + 2^2\], with a^2 - 2ab + b^2 annotated in red above it.], alt: "A mathematical expression showing the expansion of a perfect square trinomial inside brackets, specifically 4y[(5x)^2 - 2 * 5x * 2 + 2^2], with a^2 - 2ab + b^2 annotated in red above it.", caption: none)],
  [Factor.], [#figure(figph[A mathematical expression showing 4y multiplied by the quantity (5x minus 2) squared, written as 4y(5x - 2)×2.], alt: "A mathematical expression showing 4y multiplied by the quantity (5x minus 2) squared, written as 4y(5x - 2)×2.", caption: none)],
))

Remember: Keep the factor 4#emph[y] in the final product.

Check: #linebreak()  #linebreak() #math.equation(block: false, alt: "4 y open parenthesis 5 x minus 2 close parenthesis squared; 4 y open bracket open parenthesis 5 x close parenthesis squared minus 2 times 5 x times 2 plus 2 squared close bracket; 4 y open parenthesis 25 x squared minus 20 x plus 4 close parenthesis; 100 x squared y minus 80 x y plus 16 y ✓")[$#h(4em) 4 y attach(( 5 x − 2 ), t: 2) \ 4 y [ attach(( 5 x ), t: 2) − 2 · 5 x · 2 + 2^(2) ] \ 4 y ( 25 x^(2) − 20 x + 4 ) \ 100 x^(2) y − 80 x y + 16 y ✓$]
]
]

#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:

#figure(figph[We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.], alt: "We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.", caption: none)

A 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[a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.], alt: "a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are 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 5")[How to Factor a Binomial Using the Difference of Squares][
Factor: #math.equation(block: false, alt: "64 y squared minus 1.")[$64 y^(2) − 1 .$]

#solutionbox[
#figure(figph[Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.], alt: "Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.", caption: none)

#figure(figph[Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.], alt: "Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.", caption: none)

#figure(figph[Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.], alt: "Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.", caption: none)

#figure(figph[Step 4 is to check. We multiply to get the original binomial], alt: "Step 4 is to check. We multiply to get the original binomial", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "121 m squared minus 1.")[$121 m^(2) − 1 .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 11 m minus 1 close parenthesis open parenthesis 11 m plus 1 close parenthesis")[$( 11 m − 1 ) ( 11 m + 1 )$]
]
]

#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")[$( 9 y − 1 ) ( 9 y + 1 )$]
]
]

#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; 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.")[$"Step 1." & "Does the binomial fit the pattern?" &  &  & #h(4em) a^(2) − b^(2) \  & "Is this a difference?" &  &  & #h(4em) "____" − "____" \  & "Are the first and last terms perfect squares?" &  &  &  \ "Step 2." & "Write them as squares." &  &  & #h(4em) attach(( a ), t: 2) − attach(( b ), t: 2) \ "Step 3." & "Write the product of conjugates." &  &  & #h(4em) ( a − b ) ( a + b ) \ "Step 4." & "Check by multiplying." &  &  &$]
]

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!

The next example shows variables in both terms.

#examplebox("Example 6")[][
Factor: #math.equation(block: false, alt: "144 x squared minus 49 y squared .")[$144 x^(2) − 49 y^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "144 x squared minus 49 y squared")[$144 x^(2) − 49 y^(2)$]]),
  [Is this a difference of squares? Yes.], [#math.equation(block: false, alt: "open parenthesis 12 x close parenthesis squared minus open parenthesis 7 y close parenthesis squared")[$attach(( 12 x ), t: 2) − attach(( 7 y ), t: 2)$]],
  [Factor as the product of conjugates.], [#math.equation(block: false, alt: "open parenthesis 12 x minus 7 y close parenthesis open parenthesis 12 x plus 7 y close parenthesis")[$( 12 x − 7 y ) ( 12 x + 7 y )$]],
  [Check by multiplying. #linebreak() 
#math.equation(block: false, alt: "open parenthesis 12 x minus 7 y close parenthesis open parenthesis 12 x plus 7 y close parenthesis; 144 x squared minus 49 y squared ✓")[$#h(4em) ( 12 x − 7 y ) ( 12 x + 7 y ) \ 144 x^(2) − 49 y^(2) ✓$]], [],
))
]
]

#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: "121 p squared minus 9 q squared .")[$121 p^(2) − 9 q^(2) .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 11 p minus 3 q close parenthesis open parenthesis 11 p plus 3 q close parenthesis")[$( 11 p − 3 q ) ( 11 p + 3 q )$]
]
]

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.

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

#examplebox("Example 7")[][
Factor: #math.equation(block: false, alt: "48 x to the power 4 y squared minus 243 y squared .")[$48 x^(4) y^(2) − 243 y^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "48 x to the power 4 y squared minus 243 y squared")[$48 x^(4) y^(2) − 243 y^(2)$]]),
  [Is there a GCF? Yes, #math.equation(block: false, alt: "3 y squared")[$3 y^(2)$]—factor it out!], [#math.equation(block: false, alt: "3 y squared open parenthesis 16 x to the power 4 minus 81 close parenthesis")[$3 y^(2) ( 16 x^(4) − 81 )$]],
  [Is the binomial a difference of squares? Yes.], [#math.equation(block: false, alt: "3 y squared open parenthesis open parenthesis 4 x squared close parenthesis squared minus open parenthesis 9 close parenthesis squared close parenthesis")[$3 y^(2) ( attach(( 4 x^(2) ), t: 2) − attach(( 9 ), t: 2) )$]],
  [Factor as a product of conjugates.], [#math.equation(block: false, alt: "3 y squared open parenthesis 4 x squared minus 9 close parenthesis open parenthesis 4 x squared plus 9 close parenthesis")[$3 y^(2) ( 4 x^(2) − 9 ) ( 4 x^(2) + 9 )$]],
  [Notice the first binomial is also a difference of squares!], [#math.equation(block: false, alt: "3 y squared open parenthesis open parenthesis 2 x close parenthesis squared minus open parenthesis 3 close parenthesis squared close parenthesis open parenthesis 4 x squared plus 9 close parenthesis")[$3 y^(2) ( attach(( 2 x ), t: 2) − attach(( 3 ), t: 2) ) ( 4 x^(2) + 9 )$]],
  [Factor it as the product of conjugates.], [#math.equation(block: false, alt: "3 y squared open parenthesis 2 x minus 3 close parenthesis open parenthesis 2 x plus 3 close parenthesis open parenthesis 4 x squared plus 9 close parenthesis")[$3 y^(2) ( 2 x − 3 ) ( 2 x + 3 ) ( 4 x^(2) + 9 )$]],
  [The last factor, the sum of squares, cannot be factored.], [],
  [Check by multiplying:], [],
  [#math.equation(block: false, alt: "3 y squared open parenthesis 2 x minus 3 close parenthesis open parenthesis 2 x plus 3 close parenthesis open parenthesis 4 x squared plus 9 close parenthesis")[$3 y^(2) ( 2 x − 3 ) ( 2 x + 3 ) ( 4 x^(2) + 9 )$]], [],
  [#math.equation(block: false, alt: "3 y squared open parenthesis 4 x squared minus 9 close parenthesis open parenthesis 4 x squared plus 9 close parenthesis")[$3 y^(2) ( 4 x^(2) − 9 ) ( 4 x^(2) + 9 )$]], [],
  [#math.equation(block: false, alt: "3 y squared open parenthesis 16 x to the power 4 minus 81 close parenthesis")[$3 y^(2) ( 16 x^(4) − 81 )$]], [],
  [#math.equation(block: false, alt: "48 x to the power 4 y squared minus 243 y squared ✓")[$48 x^(4) y^(2) − 243 y^(2) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "2 x to the power 4 y squared minus 32 y squared .")[$2 x^(4) y^(2) − 32 y^(2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "7 a to the power 4 c squared minus 7 b to the power 4 c squared .")[$7 a^(4) c^(2) − 7 b^(4) c^(2) .$]

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

The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.

#examplebox("Example 8")[][
Factor: #math.equation(block: false, alt: "x squared minus 6 x plus 9 minus y squared .")[$x^(2) − 6 x + 9 − y^(2) .$]

#solutionbox[
Notice that the first three terms form a perfect square trinomial.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays x squared minus 6x plus 9 minus y squared.], alt: "A mathematical expression displays x squared minus 6x plus 9 minus y squared.", caption: none)]),
  [Factor by grouping the first three terms.], [#figure(figph[A mathematical expression showing x^2 - 6x + 9 - y^2, with a brace underneath the first three terms (x^2 - 6x + 9) indicating they form a group.], alt: "A mathematical expression showing x^2 - 6x + 9 - y^2, with a brace underneath the first three terms (x^2 - 6x + 9) indicating they form a group.", caption: none)],
  [Use the perfect square trinomial pattern.], [#figure(figph[The mathematical expression (x-3)^2 - y^2 is displayed, representing the difference of two squares with a binomial as the first squared term.], alt: "The mathematical expression (x-3)^2 - y^2 is displayed, representing the difference of two squares with a binomial as the first squared term.", caption: none)],
  [Is this a difference of squares? Yes.], [],
  [Yes—write them as squares.], [#figure(figph[Two mathematical expressions are displayed: a^2 - b^2 in red text, and (x-3)^2 - y^2 in black text, both representing the difference of squares.], alt: "Two mathematical expressions are displayed: a^2 - b^2 in red text, and (x-3)^2 - y^2 in black text, both representing the difference of squares.", caption: none)],
  [Factor as the product of conjugates.], [#figure(figph[The image shows an algebraic expression \`((x-3)-y)((x-3)+y)\`, which is an application of the difference of squares formula (A-B)(A+B).], alt: "The image shows an algebraic expression `((x-3)-y)((x-3)+y)`, which is an application of the difference of squares formula (A-B)(A+B).", caption: none)],
  [], [#figure(figph[A mathematical expression showing the product of two binomials: (x - 3 - y)(x - 3 + y). This is an example of the difference of squares formula.], alt: "A mathematical expression showing the product of two binomials: (x - 3 - y)(x - 3 + y). This is an example of the difference of squares formula.", caption: none)],
))

You may want to rewrite the solution as #math.equation(block: false, alt: "open parenthesis x minus y minus 3 close parenthesis open parenthesis x plus y minus 3 close parenthesis .")[$( x − y − 3 ) ( x + y − 3 ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "x squared minus 10 x plus 25 minus y squared .")[$x^(2) − 10 x + 25 − y^(2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "x squared plus 6 x plus 9 minus 4 y squared .")[$x^(2) + 6 x + 9 − 4 y^(2) .$]

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


=== 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 algebraic identity for the sum of cubes: (a + b)(a^2 - ab + b^2).], alt: "The algebraic identity for the sum of cubes: (a + b)(a^2 - ab + b^2).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression showing the sum of two terms: a multiplied by (a squared minus ab plus b squared) plus b multiplied by (a squared minus ab plus b squared). This simplifies to a cubed plus b cubed.], alt: "A mathematical expression showing the sum of two terms: a multiplied by (a squared minus ab plus b squared) plus b multiplied by (a squared minus ab plus b squared). This simplifies to a cubed plus b cubed.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression: a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3. This expression simplifies to a^3 + b^3.], alt: "A mathematical expression: a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3. This expression simplifies to a^3 + b^3.", caption: none)],
  [Combine like terms.], [#figure(figph[The mathematical expression a^3 + b^3 is displayed, representing the sum of two cubes.], alt: "The mathematical expression a^3 + b^3 is displayed, representing the sum of two cubes.", caption: none)],
))

#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[a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.], alt: "a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.", caption: none)

The trinomial factor in the sum and difference of cubes pattern cannot be factored.

It will 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 9")[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[Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are], alt: "Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are", caption: none)

#figure(figph[Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.], alt: "Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.", caption: none)

#figure(figph[Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.], alt: "Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.", caption: none)

#figure(figph[Step 4 is to simplify inside the parentheses. It is already simplified], alt: "Step 4 is to simplify inside the parentheses. It is already simplified", caption: none)

#figure(figph[Step 5 is to check by multiplying the factors.], alt: "Step 5 is to check by multiplying the factors.", 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.]

+ Does the binomial fit the sum or difference of cubes pattern? #linebreak() Is it a sum or difference? #linebreak() 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 10")[][
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[The mathematical expression 27u^3 - 125v^3 is displayed in black text on a white background, representing the difference of two cubes.], alt: "The mathematical expression 27u^3 - 125v^3 is displayed in black text on a white background, representing the difference of two cubes.", caption: none)]),
  [This binomial is a difference. The first and last #linebreak() terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[A mathematical image showing the difference of cubes identity: a³ - b³ in red, followed by a specific application of the formula: (3u)³ - (5v)³.], alt: "A mathematical image showing the difference of cubes identity: a³ - b³ in red, followed by a specific application of the formula: (3u)³ - (5v)³.", caption: none)],
  [Use the difference of cubes pattern.], [#figure(figph[An image displaying the algebraic identity for the difference of cubes, (a - b)(a^2 + ab + b^2), with a specific example below it where a=3u and b=5v.], alt: "An image displaying the algebraic identity for the difference of cubes, (a - b)(a^2 + ab + b^2), with a specific example below it where a=3u and b=5v.", caption: none)],
  [Simplify.], [#figure(figph[Two lines of algebraic expressions, illustrating the difference of cubes identity: (a - b)(a^2 + ab + b^2) and (3u - 5v)(9u^2 + 15uv + 25v^2).], alt: "Two lines of algebraic expressions, illustrating the difference of cubes identity: (a - b)(a^2 + ab + b^2) and (3u - 5v)(9u^2 + 15uv + 25v^2).", 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 11")[][
Factor: #math.equation(block: false, alt: "6 x cubed y plus 48 y to the power 4.")[$6 x^(3) y + 48 y^(4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression 6x^3y + 48y^4 is displayed in black text on a white background, featuring variables x and y raised to powers, coefficients, and an addition operator.], alt: "A mathematical expression 6x^3y + 48y^4 is displayed in black text on a white background, featuring variables x and y raised to powers, coefficients, and an addition operator.", caption: none)]),
  [Factor the common factor.], [#figure(figph[A mathematical expression is displayed, which reads as 6y multiplied by the sum of x cubed and 8y cubed, written as 6y(x^3 + 8y^3).], alt: "A mathematical expression is displayed, which reads as 6y multiplied by the sum of x cubed and 8y cubed, written as 6y(x^3 + 8y^3).", caption: none)],
  [This binomial is a sum The first and last #linebreak() terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[A mathematical expression 6y(x^3 + (2y)^3), with a hint above in red indicating the sum of cubes formula, a^3 + b^3.], alt: "A mathematical expression 6y(x^3 + (2y)^3), with a hint above in red indicating the sum of cubes formula, a^3 + b^3.", caption: none)],
  [Use the sum of cubes pattern.], [#figure(figph[A mathematical expression featuring 6y multiplied by two parenthetical terms: (x + 2y) and (x^2 - x \* 2y + (2y)^2], alt: "A mathematical expression featuring 6y multiplied by two parenthetical terms: (x + 2y) and (x^2 - x * 2y + (2y)^2", caption: none)],
  [Simplify.], [#figure(figph[The algebraic expression 6y(x + 2y)(x^2 - 2xy + 4y^2), which is a factored form related to the sum of cubes, simplifying to 6y(x^3 + 8y^3).], alt: "The algebraic expression 6y(x + 2y)(x^2 - 2xy + 4y^2), which is a factored form related to the sum of cubes, simplifying to 6y(x^3 + 8y^3).", caption: none)],
))

Check:

To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by #math.equation(block: false, alt: "6 y .")[$6 y .$] We’ll leave the multiplication for you.
]
]

#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")[$2 ( 6 c + 7 d ) ( 36 c^(2) − 42 c d + 49 d^(2) )$]
]
]

The first term in the next example is a binomial cubed.

#examplebox("Example 12")[][
Factor: #math.equation(block: false, alt: "open parenthesis x plus 5 close parenthesis cubed minus 64 x cubed .")[$attach(( x + 5 ), t: 3) − 64 x^(3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the mathematical expression (x+5)² - 64x³.], alt: "The image displays the mathematical expression (x+5)² - 64x³.", caption: none)]),
  [This binomial is a difference. The first and #linebreak() last terms are perfect cubes.], [],
  [Write the terms as cubes.], [#figure(figph[The image displays the difference of cubes formula a^3 - b^3 in red, above an example expression (x + 5)^3 - (4x)^3, demonstrating its algebraic application.], alt: "The image displays the difference of cubes formula a^3 - b^3 in red, above an example expression (x + 5)^3 - (4x)^3, demonstrating its algebraic application.", caption: none)],
  [Use the difference of cubes pattern.], [#figure(figph[A mathematical expression illustrating the difference of cubes factorization (a-b)(a^2+ab+b^2), with 'a' representing (x+5) and 'b' representing 4x.], alt: "A mathematical expression illustrating the difference of cubes factorization (a-b)(a^2+ab+b^2), with 'a' representing (x+5) and 'b' representing 4x.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression featuring the product of two polynomials: (x + 5 - 4x) multiplied by (x^2 + 10x + 25 + 4x^2 + 20x + 16x^2).], alt: "A mathematical expression featuring the product of two polynomials: (x + 5 - 4x) multiplied by (x^2 + 10x + 25 + 4x^2 + 20x + 16x^2).", caption: none)],
  [], [#figure(figph[A mathematical expression showing the product of two polynomials: (-3x + 5) and (21x^2 + 30x + 25).], alt: "A mathematical expression showing the product of two polynomials: (-3x + 5) and (21x^2 + 30x + 25).", 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: "open parenthesis y plus 1 close parenthesis cubed minus 27 y cubed .")[$attach(( y + 1 ), t: 3) − 27 y^(3) .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −2 y plus 1 close parenthesis open parenthesis 13 y squared plus 5 y plus 1 close parenthesis")[$( −2 y + 1 ) ( 13 y^(2) + 5 y + 1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "open parenthesis n plus 3 close parenthesis cubed minus 125 n cubed .")[$attach(( n + 3 ), t: 3) − 125 n^(3) .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −4 n plus 3 close parenthesis open parenthesis 31 n squared plus 21 n plus 9 close parenthesis")[$( −4 n + 3 ) ( 31 n^(2) + 21 n + 9 )$]
]
]

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

- #link("https://openstax.org/l/37BinomCubes")[Factoring Binomials-Cubes \#2]
]


=== Key Concepts

- #strong[Perfect Square Trinomials Pattern:] If #emph[a] and #emph[b] are real numbers, #linebreak() 
#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")[$a^(2) + 2 a b + b^(2) = attach(( a + b ), t: 2) \ a^(2) − 2 a b + b^(2) = attach(( a − b ), t: 2)$]
- #strong[How to 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 squ")[$"Step 1." & "Does the trinomial fit the pattern?" &  &  & #h(4em) a^(2) + 2 a b + b^(2) &  &  & a^(2) − 2 a b + b^(2) \  & "Is the first term a perfect square?" &  &  & #h(4em) attach(( a ), t: 2) &  &  & attach(( a ), t: 2) \  & "Write it as a square." &  &  &  &  &  &  \  & "Is the last term a perfect square?" &  &  & #h(4em) attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) &  &  & attach(( a ), t: 2) #h(4.5em) attach(( b ), t: 2) \  & "Write it as a square." &  &  &  &  &  &  \  & "Check the middle term. Is it" #h(0.2em) 2 a b ? &  &  & #h(4em) attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) &  &  & attach(( a ), t: 2) "↘" limits(zws)_(2 · a · b) "↙" attach(( b ), t: 2) \ "Step 2." & "Write the square of the binomial." &  &  & #h(4em) attach(( a + b ), t: 2) &  &  & attach(( a − b ), t: 2) \ "Step 3." & "Check by multiplying." &  &  &  &  &  &$]
- #strong[Difference of Squares Pattern:] If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, #linebreak() 
#figure(figph[a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.], alt: "a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.", caption: none)
- #strong[How to 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.")[$"Step 1." & "Does the binomial fit the pattern?" &  &  & #h(4em) a^(2) − b^(2) \  & "Is this a difference?" &  &  & #h(4em) "____" − "____" \  & "Are the first and last terms perfect squares?" &  &  &  \ "Step 2." & "Write them as squares." &  &  & #h(4em) attach(( a ), t: 2) − attach(( b ), t: 2) \ "Step 3." & "Write the product of conjugates." &  &  & #h(4em) ( a − b ) ( a + b ) \ "Step 4." & "Check by multiplying." &  &  &$]
- #strong[Sum and Difference of Cubes Pattern] #linebreak() 
#math.equation(block: false, 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")[$a^(3) + b^(3) = ( a + b ) ( a^(2) − a b + b^(2) ) \ a^(3) − b^(3) = ( a − b ) ( a^(2) + a b + b^(2) )$]
- #strong[How to factor the sum or difference of cubes.]
+ Does the binomial fit the sum or difference of cubes pattern? #linebreak() Is it a sum or difference? #linebreak() 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 completely using the perfect square trinomials pattern.

#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 plus 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 plus 2 y close parenthesis squared")[$attach(( 7 x + 2 y ), t: 2)$]
]

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

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

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

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

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

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

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

#math.equation(block: true, alt: "75 u to the power 4 minus 30 u cubed v plus 3 u squared v squared")[$75 u^(4) − 30 u^(3) v + 3 u^(2) v^(2)$]

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

#math.equation(block: true, alt: "90 p to the power 4 plus 300 p cubed q plus 250 p squared q squared")[$90 p^(4) + 300 p^(3) q + 250 p^(2) q^(2)$]

#strong[Factor Differences of Squares]

In the following exercises, factor completely using the difference of squares pattern, if possible.

#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: "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: "6 p squared q squared minus 54 p squared")[$6 p^(2) q^(2) − 54 p^(2)$]

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

#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")[$20 b^(2) + 140$]

#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: "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: "162 a to the power 4 b squared minus 32 b squared")[$162 a^(4) b^(2) − 32 b^(2)$]

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

#math.equation(block: true, alt: "48 m to the power 4 n squared minus 243 n squared")[$48 m^(4) n^(2) − 243 n^(2)$]

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

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

#math.equation(block: true, alt: "p squared plus 14 p plus 49 minus q squared")[$p^(2) + 14 p + 49 − q^(2)$]

#math.equation(block: true, alt: "a squared plus 6 a plus 9 minus 9 b squared")[$a^(2) + 6 a + 9 − 9 b^(2)$]

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

#math.equation(block: true, alt: "m squared minus 6 m plus 9 minus 16 n squared")[$m^(2) − 6 m + 9 − 16 n^(2)$]

#strong[Factor Sums and Differences of Cubes]

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

#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 to the power 6 plus 512")[$n^(6) + 512$]

#math.equation(block: true, alt: "z to the power 6 minus 27")[$z^(6) − 27$]

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

#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: "216 a cubed plus 125 b cubed")[$216 a^(3) + 125 b^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 6 a plus 5 b close parenthesis open parenthesis 36 a squared minus 30 a b plus 25 b squared close parenthesis")[$( 6 a + 5 b ) ( 36 a^(2) − 30 a b + 25 b^(2) )$]
]

#math.equation(block: true, alt: "27 y cubed plus 8 z cubed")[$27 y^(3) + 8 z^(3)$]

#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 x squared minus 16 x squared y cubed")[$2 x^(2) − 16 x^(2) y^(3)$]

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

#math.equation(block: true, alt: "−2 x cubed y squared minus 16 y to the power 5")[$−2 x^(3) y^(2) − 16 y^(5)$]

#math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis cubed plus 8 x cubed")[$attach(( x + 3 ), t: 3) + 8 x^(3)$]

#solutionbox[
#math.equation(block: true, alt: "9 open parenthesis x plus 1 close parenthesis open parenthesis x squared plus 3 close parenthesis")[$9 ( x + 1 ) ( x^(2) + 3 )$]
]

#math.equation(block: true, alt: "open parenthesis x plus 4 close parenthesis cubed minus 27 x cubed")[$attach(( x + 4 ), t: 3) − 27 x^(3)$]

#math.equation(block: true, alt: "open parenthesis y minus 5 close parenthesis cubed minus 64 y cubed")[$attach(( y − 5 ), t: 3) − 64 y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "− open parenthesis 3 y plus 5 close parenthesis open parenthesis 21 y squared minus 30 y plus 25 close parenthesis")[$"−" ( 3 y + 5 ) ( 21 y^(2) − 30 y + 25 )$]
]

#math.equation(block: true, alt: "open parenthesis y minus 5 close parenthesis cubed plus 125 y cubed")[$attach(( y − 5 ), t: 3) + 125 y^(3)$]

#strong[Mixed Practice]

In the following exercises, factor completely.

#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")[$48 q^(3) − 24 q^(2) + 3 q$]

#math.equation(block: true, alt: "x squared minus 10 x plus 25 minus y squared")[$x^(2) − 10 x + 25 − y^(2)$]

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

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

#math.equation(block: true, alt: "open parenthesis x plus 1 close parenthesis cubed plus 8 x cubed")[$attach(( x + 1 ), t: 3) + 8 x^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 x plus 1 close parenthesis open parenthesis 3 x squared plus 1 close parenthesis")[$( 3 x + 1 ) ( 3 x^(2) + 1 )$]
]

#math.equation(block: true, alt: "open parenthesis y minus 3 close parenthesis cubed minus 64 y cubed")[$attach(( y − 3 ), t: 3) − 64 y^(3)$]


==== Writing Exercises

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

#solutionbox[
Answers will 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 will 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 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.], alt: "This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?