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== 6.4#h(0.6em)Special Products

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

Simplify: ⓐ #math.equation(block: false, alt: "9 squared")[$9^(2)$] ⓑ #math.equation(block: false, alt: "open parenthesis −9 close parenthesis squared")[$attach(( −9 ), t: 2)$] ⓒ #math.equation(block: false, alt: "− 9 squared .")[$"−" 9^(2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
ⓐ #math.equation(block: false, alt: "81")[$81$] ⓑ #math.equation(block: false, alt: "81")[$81$] ⓒ #math.equation(block: false, alt: "minus 81")[$− 81$]
]
]


=== Square a Binomial Using the Binomial Squares Pattern

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the #strong[binomial] twice and using the methods of the last section, there is less work to do if you learn to use a pattern.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Let's start by looking at #math.equation(block: false, alt: "open parenthesis x plus 9 close parenthesis squared")[$attach(( x + 9 ), t: 2)$].], []),
  [What does this mean?], [#math.equation(block: false, alt: "open parenthesis x plus 9 close parenthesis squared")[$attach(( x + 9 ), t: 2)$]],
  [It means to multiply #math.equation(block: false, alt: "open parenthesis x plus 9 close parenthesis")[$( x + 9 )$] by itself.], [#math.equation(block: false, alt: "open parenthesis x plus 9 close parenthesis open parenthesis x plus 9 close parenthesis")[$( x + 9 ) ( x + 9 )$]],
  [Then, using FOIL, we get:], [#math.equation(block: false, alt: "x squared plus 9 x plus 9 x plus 81")[$x^(2) + 9 x + 9 x + 81$]],
  [Combining like terms gives:], [#math.equation(block: false, alt: "x squared plus 18 x plus 81")[$x^(2) + 18 x + 81$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Here's another one:], [#math.equation(block: false, alt: "open parenthesis y minus 7 close parenthesis squared")[$attach(( y − 7 ), t: 2)$]]),
  [Multiply #math.equation(block: false, alt: "open parenthesis y minus 7 close parenthesis")[$( y − 7 )$] by itself.], [#math.equation(block: false, alt: "open parenthesis y minus 7 close parenthesis open parenthesis y minus 7 close parenthesis")[$( y − 7 ) ( y − 7 )$]],
  [Using FOIL, we get:], [#math.equation(block: false, alt: "y squared minus 7 y minus 7 y plus 49")[$y^(2) − 7 y − 7 y + 49$]],
  [And combining like terms:], [#math.equation(block: false, alt: "y squared minus 14 y plus 49")[$y^(2) − 14 y + 49$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([And one more:], [#math.equation(block: false, alt: "open parenthesis 2 x plus 3 close parenthesis squared")[$attach(( 2 x + 3 ), t: 2)$]]),
  [Multiply.], [#math.equation(block: false, alt: "open parenthesis 2 x plus 3 close parenthesis open parenthesis 2 x plus 3 close parenthesis")[$( 2 x + 3 ) ( 2 x + 3 )$]],
  [Use FOIL:], [#math.equation(block: false, alt: "4 x squared plus 6 x plus 6 x plus 9")[$4 x^(2) + 6 x + 6 x + 9$]],
  [Combine like terms.], [#math.equation(block: false, alt: "4 x squared plus 12 x plus 9")[$4 x^(2) + 12 x + 9$]],
))

Look at these results. Do you see any patterns?

What about the number of terms? In each example we squared a binomial and the result was a #strong[trinomial].

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals ____ plus ____ plus ____")[$attach(( a + b ), t: 2) = "____" + "____" + "____"$]Now look at the #strong[#emph[first term]] in each result. Where did it come from?

#figure(figph[This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.], alt: "This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.", caption: none)

The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus ____ plus ____")[$attach(( a + b ), t: 2) = a^(2) + "____" + "____"$]To get the #strong[#emph[first term]] of the product, #strong[#emph[square the first term]].

Where did the #strong[#emph[last term]] come from? Look at the examples and find the pattern.

The last term is the product of the last terms, which is the square of the last term.

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals ____ plus ____ plus b squared")[$attach(( a + b ), t: 2) = "____" + "____" + b^(2)$]#emph[To get the #strong[last term] of the product, #strong[square the last term]].

Finally, look at the #strong[#emph[middle term]]. Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals ____ plus 2 a b plus ____; open parenthesis a minus b close parenthesis squared equals ____ minus 2 a b plus ____")[$attach(( a + b ), t: 2) = "____" + 2 a b + "____" \ attach(( a − b ), t: 2) = "____" − 2 a b + "____"$]#emph[To get the #strong[middle term] of the product, #strong[multiply the terms and double their product]].

Putting it all together:

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

If #math.equation(block: false, alt: "a and b")[$a #h(0.2em) "and" #h(0.2em) b$] 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")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) \ attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2)$]#figure(figph[The image illustrates the binomial squares pattern. It defines a as the first term and b as the last term.], alt: "The image illustrates the binomial squares pattern. It defines a as the first term and b as the last term.", caption: none)
]

To square a binomial:

- square the first term
- square the last term
- double their product

A number example helps verify the pattern.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 10 plus 4 close parenthesis squared")[$attach(( 10 + 4 ), t: 2)$]]),
  [Square the first term.], [#math.equation(block: false, alt: "10 squared plus ___ plus")[$10^(2) + "___" +$]],
  [Square the last term.], [#math.equation(block: false, alt: "10 squared plus ___ plus 4 squared")[$10^(2) + "___" + 4^(2)$]],
  [Double their product.], [#math.equation(block: false, alt: "10 squared plus 2 times 10 times 4 plus 4 squared")[$10^(2) + 2 · 10 · 4 + 4^(2)$]],
  [Simplify.], [#math.equation(block: false, alt: "100 plus 80 plus 16")[$100 + 80 + 16$]],
  [Simplify.], [#math.equation(block: false, alt: "196")[$196$]],
))

To multiply #math.equation(block: false, alt: "open parenthesis 10 plus 4 close parenthesis squared")[$attach(( 10 + 4 ), t: 2)$] usually you’d follow the Order of Operations.

#math.equation(block: true, alt: "open parenthesis 10 plus 4 close parenthesis squared; open parenthesis 14 close parenthesis squared; 196")[$attach(( 10 + 4 ), t: 2) \ attach(( 14 ), t: 2) \ 196$]The pattern works!

#examplebox("Example 1")[][
Multiply: #math.equation(block: false, alt: "open parenthesis x plus 5 close parenthesis squared .")[$attach(( x + 5 ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.], alt: "x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.", caption: none)]),
  [Square the first term.], [#figure(figph[x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.], alt: "x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.", caption: none)],
  [Square the last term.], [#figure(figph[x squared plus blank plus 5 squared.], alt: "x squared plus blank plus 5 squared.", caption: none)],
  [Double the product.], [#figure(figph[x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.], alt: "x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[x squared plus 10 x plus 25.], alt: "x squared plus 10 x plus 25.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis x plus 9 close parenthesis squared .")[$attach(( x + 9 ), t: 2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis y plus 11 close parenthesis squared .")[$attach(( y + 11 ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "y squared plus 22 y plus 121")[$y^(2) + 22 y + 121$]
]
]

#examplebox("Example 2")[][
Multiply: #math.equation(block: false, alt: "open parenthesis y minus 3 close parenthesis squared .")[$attach(( y − 3 ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.], alt: "y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.", caption: none)]),
  [Square the first term.], [#figure(figph[y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.], alt: "y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.", caption: none)],
  [Square the last term.], [#figure(figph[y squared minus blank plus 3 squared.], alt: "y squared minus blank plus 3 squared.", caption: none)],
  [Double the product.], [#figure(figph[y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.], alt: "y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[y squared minus 6 y plus 9.], alt: "y squared minus 6 y plus 9.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis x minus 9 close parenthesis squared .")[$attach(( x − 9 ), t: 2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis p minus 13 close parenthesis squared .")[$attach(( p − 13 ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "p squared minus 26 p plus 169")[$p^(2) − 26 p + 169$]
]
]

#examplebox("Example 3")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 4 x plus 6 close parenthesis squared .")[$attach(( 4 x + 6 ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.], alt: "4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.", caption: none)]),
  [Use the pattern.], [#figure(figph[4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.], alt: "4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[16 x squared plus 48 x plus 36.], alt: "16 x squared plus 48 x plus 36.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 6 x plus 3 close parenthesis squared .")[$attach(( 6 x + 3 ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "36 x squared plus 36 x plus 9")[$36 x^(2) + 36 x + 9$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 4 x plus 9 close parenthesis squared .")[$attach(( 4 x + 9 ), t: 2) .$]

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

#examplebox("Example 4")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 2 x minus 3 y close parenthesis squared .")[$attach(( 2 x − 3 y ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.], alt: "contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.", caption: none)]),
  [Use the pattern.], [#figure(figph[2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.], alt: "2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[4 x squared minus 12 x y plus 9 y squared.], alt: "4 x squared minus 12 x y plus 9 y squared.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 2 c minus d close parenthesis squared .")[$attach(( 2 c − d ), t: 2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 4 x minus 5 y close parenthesis squared .")[$attach(( 4 x − 5 y ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "16 x squared minus 40 x y plus 25 y squared")[$16 x^(2) − 40 x y + 25 y^(2)$]
]
]

#examplebox("Example 5")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 4 u cubed plus 1 close parenthesis squared .")[$attach(( 4 u^(3) + 1 ), t: 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.], alt: "4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.", caption: none)]),
  [Use the pattern.], [#figure(figph[4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.], alt: "4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[16 u to the sixth power plus 18 u cubed plus 1.], alt: "16 u to the sixth power plus 18 u cubed plus 1.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 2 x squared plus 1 close parenthesis squared .")[$attach(( 2 x^(2) + 1 ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "4 x to the power 4 plus 4 x squared plus 1")[$4 x^(4) + 4 x^(2) + 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 3 y cubed plus 2 close parenthesis squared .")[$attach(( 3 y^(3) + 2 ), t: 2) .$]

#solutionbox[
#math.equation(block: true, alt: "9 y to the power 6 plus 12 y cubed plus 4")[$9 y^(6) + 12 y^(3) + 4$]
]
]


=== Multiply Conjugates Using the Product of Conjugates Pattern

We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.

What do you notice about these pairs of binomials?

#math.equation(block: true, alt: "open parenthesis x minus 9 close parenthesis open parenthesis x plus 9 close parenthesis, open parenthesis y minus 8 close parenthesis open parenthesis y plus 8 close parenthesis, open parenthesis 2 x minus 5 close parenthesis open parenthesis 2 x plus 5 close parenthesis")[$( x − 9 ) ( x + 9 ) &  &  & #h(4em) ( y − 8 ) ( y + 8 ) &  &  & #h(4em) ( 2 x − 5 ) ( 2 x + 5 )$]Look at the first term of each #strong[binomial] in each pair.

#figure(figph[This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses], alt: "This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses", caption: none)

#emph[Notice the first terms are the same in each pair.]

Look at the last terms of each binomial in each pair.

#figure(figph[This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.], alt: "This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.", caption: none)

#emph[Notice the last terms are the same in each pair.]

#emph[Notice how each pair has one sum and one difference.]

#figure(figph[This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word “difference”. Below x plus 9 is the word “sum”. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word “difference”. Below y plus 8 is the word “sum”. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word “difference” and below 2x plus 5 is the word “sum”.], alt: "This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word “difference”. Below x plus 9 is the word “sum”. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word “difference”. Below y plus 8 is the word “sum”. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word “difference” and below 2x plus 5 is the word “sum”.", caption: none)

A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a #emph[conjugate pair] and is of the form #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis , open parenthesis a plus b close parenthesis")[$( a − b ) , ( a + b )$].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Conjugate Pair]

A #strong[conjugate pair] is two binomials of the form

#math.equation(block: true, alt: "open parenthesis a minus b close parenthesis , open parenthesis a plus b close parenthesis .")[$( a − b ) , ( a + b ) .$]The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.
]

There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.

Let’s look for the pattern by using FOIL to multiply some conjugate pairs.

#math.equation(block: true, alt: "open parenthesis x minus 9 close parenthesis open parenthesis x plus 9 close parenthesis, open parenthesis y minus 8 close parenthesis open parenthesis y plus 8 close parenthesis, open parenthesis 2 x minus 5 close parenthesis open parenthesis 2 x plus 5 close parenthesis; x squared plus 9 x minus 9 x minus 81, y squared plus 8 y minus 8 y minus 64, 4 x squared plus 10 x minus 10 x minus 25; x squared minus 81, y squared minus 64, 4 x squared minus 25")[$( x − 9 ) ( x + 9 ) &  &  & #h(4em) ( y − 8 ) ( y + 8 ) &  &  & #h(4em) ( 2 x − 5 ) ( 2 x + 5 ) \ x^(2) + 9 x − 9 x − 81 &  &  & #h(4em) y^(2) + 8 y − 8 y − 64 &  &  & #h(4em) 4 x^(2) + 10 x − 10 x − 25 \ x^(2) − 81 &  &  & #h(4em) y^(2) − 64 &  &  & #h(4em) 4 x^(2) − 25$]#figure(figph[This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.], alt: "This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.", caption: none)

Each #strong[first term] is the product of the first terms of the binomials, and since they are identical it is the square of the first term.

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis equals a squared minus ____; To get the first term, square the first term .")[$( a + b ) ( a − b ) = a^(2) − "____" \ "To get the" #h(0.2em) "first term, square the first term" .$]The #strong[last term] came from multiplying the last terms, the square of the last term.

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis equals a squared minus b squared; To get the last term, square the last term .")[$( a + b ) ( a − b ) = a^(2) − b^(2) \ "To get the" #h(0.2em) "last term, square the last term" .$]What do you observe about the products?

The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.

Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.

The product of conjugates is always of the form #math.equation(block: false, alt: "a squared minus b squared")[$a^(2) − b^(2)$]. This is called a difference of squares.

This leads to the pattern:

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

If #math.equation(block: false, alt: "a and b")[$a #h(0.2em) "and" #h(0.2em) b$] are real numbers,

#figure(figph[This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled “conjugates”, the a squared and b squared are labeled squares and the minus sign between the squares is labeled “difference”. Therefore, the product of two conjugates is called a difference of squares.], alt: "This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled “conjugates”, the a squared and b squared are labeled squares and the minus sign between the squares is labeled “difference”. Therefore, the product of two conjugates is called a difference of squares.", caption: none)

The product is called a difference of squares.

To multiply conjugates, square the first term, square the last term, and write the product as a difference of squares.
]

Let’s test this pattern with a numerical example.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 10 minus 2 close parenthesis open parenthesis 10 plus 2 close parenthesis")[$( 10 − 2 ) ( 10 + 2 )$]]),
  [It is the product of conjudgates, so the result will be the difference of two squares.], [#math.equation(block: false, alt: "____ minus ____")[$"____" − "____"$]],
  [Square the first term.], [#math.equation(block: false, alt: "10 squared minus ____")[$10^(2) − "____"$]],
  [Square the last term.], [#math.equation(block: false, alt: "10 squared minus 2 squared")[$10^(2) − 2^(2)$]],
  [Simplify.], [#math.equation(block: false, alt: "100 minus 4")[$100 − 4$]],
  [Simplify.], [#math.equation(block: false, alt: "96")[$96$]],
  [What do you get using the order of operations?], [],
  [], [#math.equation(block: false, alt: "open parenthesis 10 minus 2 close parenthesis open parenthesis 10 plus 2 close parenthesis; open parenthesis 8 close parenthesis open parenthesis 12 close parenthesis; 96")[$( 10 − 2 ) ( 10 + 2 ) \ ( 8 ) ( 12 ) \ 96$]],
))

Notice, the result is the same!

#examplebox("Example 6")[][
Multiply: #math.equation(block: false, alt: "open parenthesis x minus 8 close parenthesis open parenthesis x plus 8 close parenthesis .")[$( x − 8 ) ( x + 8 ) .$]

#solutionbox[
First, recognize this as a product of conjugates. The binomials have the same first terms, and the same last terms, and one binomial is a sum and the other is a difference.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([It fits the pattern.], [#figure(figph[The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.], alt: "The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.", caption: none)]),
  [Square the first term, #emph[x].], [#figure(figph[x squared minus blank. Above this is the general form a squared minus b squared.], alt: "x squared minus blank. Above this is the general form a squared minus b squared.", caption: none)],
  [Square the last term, 8.], [#figure(figph[x squared minus 8 squared.], alt: "x squared minus 8 squared.", caption: none)],
  [The product is a difference of squares.], [#figure(figph[Two algebraic expressions are shown: the difference of squares formula, a×2 - b×2, in red, and a specific example, x×2 - 64, in black.], alt: "Two algebraic expressions are shown: the difference of squares formula, a×2 - b×2, in red, and a specific example, x×2 - 64, in black.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis .")[$( x − 5 ) ( x + 5 ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis w minus 3 close parenthesis open parenthesis w plus 3 close parenthesis .")[$( w − 3 ) ( w + 3 ) .$]

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

#examplebox("Example 7")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 2 x plus 5 close parenthesis open parenthesis 2 x minus 5 close parenthesis .")[$( 2 x + 5 ) ( 2 x − 5 ) .$]

#solutionbox[
Are the binomials conjugates?

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([It is the product of conjugates.], [#figure(figph[The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.], alt: "The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.", caption: none)]),
  [Square the first term, 2#emph[x].], [#figure(figph[2 x squared minus blank. Above this is the general form a squared minus b squared.], alt: "2 x squared minus blank. Above this is the general form a squared minus b squared.", caption: none)],
  [Square the last term, 5.], [#figure(figph[2 x squared minus 5 squared.], alt: "2 x squared minus 5 squared.", caption: none)],
  [Simplify. The product is a difference of squares.], [#figure(figph[4 x squared minus 25.], alt: "4 x squared minus 25.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 6 x plus 5 close parenthesis open parenthesis 6 x minus 5 close parenthesis .")[$( 6 x + 5 ) ( 6 x − 5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "36 x squared minus 25")[$36 x^(2) − 25$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 2 x plus 7 close parenthesis open parenthesis 2 x minus 7 close parenthesis .")[$( 2 x + 7 ) ( 2 x − 7 ) .$]

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

The binomials in the next example may look backwards – the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.

#examplebox("Example 8")[][
Find the product: #math.equation(block: false, alt: "open parenthesis 3 plus 5 x close parenthesis open parenthesis 3 minus 5 x close parenthesis .")[$( 3 + 5 x ) ( 3 − 5 x ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([It is the product of conjugates.], [#figure(figph[The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.], alt: "The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.", caption: none)]),
  [Use the pattern.], [#figure(figph[3 squared minus 5 x squared. Above this is the general form a squared minus b squared.], alt: "3 squared minus 5 x squared. Above this is the general form a squared minus b squared.", caption: none)],
  [Simplify.], [#figure(figph[9 minus 25 x squared.], alt: "9 minus 25 x squared.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 7 plus 4 x close parenthesis open parenthesis 7 minus 4 x close parenthesis .")[$( 7 + 4 x ) ( 7 − 4 x ) .$]

#solutionbox[
#math.equation(block: true, alt: "49 minus 16 x squared")[$49 − 16 x^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 9 minus 2 y close parenthesis open parenthesis 9 plus 2 y close parenthesis .")[$( 9 − 2 y ) ( 9 + 2 y ) .$]

#solutionbox[
#math.equation(block: true, alt: "81 minus 4 y squared")[$81 − 4 y^(2)$]
]
]

Now we’ll multiply conjugates that have two variables.

#examplebox("Example 9")[][
Find the product: #math.equation(block: false, alt: "open parenthesis 5 m minus 9 n close parenthesis open parenthesis 5 m plus 9 n close parenthesis .")[$( 5 m − 9 n ) ( 5 m + 9 n ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([This fits the pattern.], [#figure(figph[5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.], alt: "5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.", caption: none)]),
  [Use the pattern.], [#figure(figph[5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.], alt: "5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.", caption: none)],
  [Simplify.], [#figure(figph[25 m squared minus 81 n squared.], alt: "25 m squared minus 81 n squared.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis 4 p minus 7 q close parenthesis open parenthesis 4 p plus 7 q close parenthesis .")[$( 4 p − 7 q ) ( 4 p + 7 q ) .$]

#solutionbox[
#math.equation(block: true, alt: "16 p squared minus 49 q squared")[$16 p^(2) − 49 q^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis 3 x minus y close parenthesis open parenthesis 3 x plus y close parenthesis .")[$( 3 x − y ) ( 3 x + y ) .$]

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

#examplebox("Example 10")[][
Find the product: #math.equation(block: false, alt: "open parenthesis c d minus 8 close parenthesis open parenthesis c d plus 8 close parenthesis .")[$( c d − 8 ) ( c d + 8 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([This fits the pattern.], [#figure(figph[The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.], alt: "The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.", caption: none)]),
  [Use the pattern.], [#figure(figph[c d squared minus 8 squared. Above this is the general form a squared minus b squared.], alt: "c d squared minus 8 squared. Above this is the general form a squared minus b squared.", caption: none)],
  [Simplify.], [#figure(figph[c squared d squared minus 64.], alt: "c squared d squared minus 64.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis x y minus 6 close parenthesis open parenthesis x y plus 6 close parenthesis .")[$( x y − 6 ) ( x y + 6 ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis a b minus 9 close parenthesis open parenthesis a b plus 9 close parenthesis .")[$( a b − 9 ) ( a b + 9 ) .$]

#solutionbox[
#math.equation(block: true, alt: "a squared b squared minus 81")[$a^(2) b^(2) − 81$]
]
]

#examplebox("Example 11")[][
Find the product: #math.equation(block: false, alt: "open parenthesis 6 u squared minus 11 v to the power 5 close parenthesis open parenthesis 6 u squared plus 11 v to the power 5 close parenthesis .")[$( 6 u^(2) − 11 v^(5) ) ( 6 u^(2) + 11 v^(5) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([This fits the pattern.], [#figure(figph[The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.], alt: "The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.", caption: none)]),
  [Use the pattern.], [#figure(figph[6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.], alt: "6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.", caption: none)],
  [Simplify.], [#figure(figph[36 u to the fourth power minus 121 v to the tenth power.], alt: "36 u to the fourth power minus 121 v to the tenth power.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis 3 x squared minus 4 y cubed close parenthesis open parenthesis 3 x squared plus 4 y cubed close parenthesis .")[$( 3 x^(2) − 4 y^(3) ) ( 3 x^(2) + 4 y^(3) ) .$]

#solutionbox[
#math.equation(block: true, alt: "9 x to the power 4 minus 16 y to the power 6")[$9 x^(4) − 16 y^(6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the product: #math.equation(block: false, alt: "open parenthesis 2 m squared minus 5 n cubed close parenthesis open parenthesis 2 m squared plus 5 n cubed close parenthesis .")[$( 2 m^(2) − 5 n^(3) ) ( 2 m^(2) + 5 n^(3) ) .$]

#solutionbox[
#math.equation(block: true, alt: "4 m to the power 4 minus 25 n to the power 6")[$4 m^(4) − 25 n^(6)$]
]
]


=== Recognize and Use the Appropriate Special Product Pattern

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Comparing the Special Product Patterns]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Binomial Squares]], [#emph[Product of Conjugates]]),
  [#math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2)$]], [#math.equation(block: false, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared")[$( a − b ) ( a + b ) = a^(2) − b^(2)$]],
  [#math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2)$]], [],
  [- Squaring a binomial], [- Multiplying conjugates],
  [- Product is a #emph[trinomial]], [- Product is a #emph[binomial]],
  [- Inner and outer terms with FOIL are #emph[the same.]], [- Inner and outer terms with FOIL are #emph[opposites.]],
  [- Middle term is #emph[double the product] of the terms.], [- There is #emph[no] middle term.],
))
]

#examplebox("Example 12")[][
Choose the appropriate pattern and use it to find the product:

ⓐ #math.equation(block: false, alt: "open parenthesis 2 x minus 3 close parenthesis open parenthesis 2 x plus 3 close parenthesis")[$( 2 x − 3 ) ( 2 x + 3 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 x minus 5 close parenthesis squared")[$attach(( 8 x − 5 ), t: 2)$] ⓒ #math.equation(block: false, alt: "open parenthesis 6 m plus 7 close parenthesis squared")[$attach(( 6 m + 7 ), t: 2)$] ⓓ #math.equation(block: false, alt: "open parenthesis 5 x minus 6 close parenthesis open parenthesis 6 x plus 5 close parenthesis")[$( 5 x − 6 ) ( 6 x + 5 )$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "open parenthesis 2 x minus 3 close parenthesis open parenthesis 2 x plus 3 close parenthesis")[$( 2 x − 3 ) ( 2 x + 3 )$] These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([This fits the pattern.], [#figure(figph[The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.], alt: "The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.", caption: none)]),
  [Use the pattern.], [#figure(figph[2 x squared minus 3 squared. Above this is the general form a squared minus b squared.], alt: "2 x squared minus 3 squared. Above this is the general form a squared minus b squared.", caption: none)],
  [Simplify.], [#figure(figph[4 x squared minus 9.], alt: "4 x squared minus 9.", caption: none)],
))
+ ⓑ #math.equation(block: false, alt: "open parenthesis 8 x minus 5 close parenthesis squared")[$attach(( 8 x − 5 ), t: 2)$] We are asked to square a binomial. It fits the #strong[binomial squares] pattern. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.], alt: "8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.", caption: none)]),
  [Use the pattern.], [#figure(figph[8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.], alt: "8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[64 x squared minus 80 x plus 25.], alt: "64 x squared minus 80 x plus 25.", caption: none)],
))
+ ⓒ #math.equation(block: false, alt: "open parenthesis 6 m plus 7 close parenthesis squared")[$attach(( 6 m + 7 ), t: 2)$] Again, we will square a binomial so we use the #strong[binomial squares] pattern. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.], alt: "6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.", caption: none)]),
  [Use the pattern.], [#figure(figph[6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.], alt: "6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.", caption: none)],
  [Simplify.], [#figure(figph[36 m squared plus 84 m plus 49.], alt: "36 m squared plus 84 m plus 49.", caption: none)],
))
+ ⓓ #math.equation(block: false, alt: "open parenthesis 5 x minus 6 close parenthesis open parenthesis 6 x plus 5 close parenthesis")[$( 5 x − 6 ) ( 6 x + 5 )$] This product does not fit the patterns, so we will use FOIL. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 5 x minus 6 close parenthesis open parenthesis 6 x plus 5 close parenthesis")[$( 5 x − 6 ) ( 6 x + 5 )$]]),
  [Use FOIL.], [#math.equation(block: false, alt: "30 x squared plus 25 x minus 36 x minus 30")[$30 x^(2) + 25 x − 36 x − 30$]],
  [Simplify.], [#math.equation(block: false, alt: "30 x squared minus 11 x minus 30")[$30 x^(2) − 11 x − 30$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Choose the appropriate pattern and use it to find the product:

ⓐ #math.equation(block: false, alt: "open parenthesis 9 b minus 2 close parenthesis open parenthesis 2 b plus 9 close parenthesis")[$( 9 b − 2 ) ( 2 b + 9 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 9 p minus 4 close parenthesis squared")[$attach(( 9 p − 4 ), t: 2)$] ⓒ #math.equation(block: false, alt: "open parenthesis 7 y plus 1 close parenthesis squared")[$attach(( 7 y + 1 ), t: 2)$] ⓓ #math.equation(block: false, alt: "open parenthesis 4 r minus 3 close parenthesis open parenthesis 4 r plus 3 close parenthesis")[$( 4 r − 3 ) ( 4 r + 3 )$]

#solutionbox[
ⓐ FOIL; #math.equation(block: false, alt: "18 b squared plus 77 b minus 18")[$18 b^(2) + 77 b − 18$] ⓑ Binomial Squares; #math.equation(block: false, alt: "81 p squared minus 72 p plus 16")[$81 p^(2) − 72 p + 16$] ⓒ Binomial Squares; #math.equation(block: false, alt: "49 y squared plus 14 y plus 1")[$49 y^(2) + 14 y + 1$] ⓓ Product of Conjugates; #math.equation(block: false, alt: "16 r squared minus 9")[$16 r^(2) − 9$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Choose the appropriate pattern and use it to find the product:

ⓐ #math.equation(block: false, alt: "open parenthesis 6 x plus 7 close parenthesis squared")[$attach(( 6 x + 7 ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 x minus 4 close parenthesis open parenthesis 3 x plus 4 close parenthesis")[$( 3 x − 4 ) ( 3 x + 4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 x minus 5 close parenthesis open parenthesis 5 x minus 2 close parenthesis")[$( 2 x − 5 ) ( 5 x − 2 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 6 n minus 1 close parenthesis squared")[$attach(( 6 n − 1 ), t: 2)$]

#solutionbox[
ⓐ Binomial Squares; #math.equation(block: false, alt: "36 x squared plus 84 x plus 49")[$36 x^(2) + 84 x + 49$] ⓑ Product of Conjugates; #math.equation(block: false, alt: "9 x squared minus 16")[$9 x^(2) − 16$] ⓒ FOIL; #math.equation(block: false, alt: "10 x squared minus 29 x plus 10")[$10 x^(2) − 29 x + 10$] ⓓ Binomial Squares; #math.equation(block: false, alt: "36 n squared minus 12 n plus 1")[$36 n^(2) − 12 n + 1$]
]
]

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

- #link("https://openstax.org/l/25Specialprod")[Special Products]
]


=== Key Concepts

- #strong[Binomial Squares Pattern]
- If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, #linebreak() 
#figure(figph[This image illustrates the algebraic identity for squaring a binomial: (a+b)^2 = a^2 + 2ab + b^2, explaining each term as '(first term)^2', '2(product of terms)', and '(last term)^2' respectively.], alt: "This image illustrates the algebraic identity for squaring a binomial: (a+b)^2 = a^2 + 2ab + b^2, explaining each term as '(first term)^2', '2(product of terms)', and '(last term)^2' respectively.", caption: none)
- #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2)$]
- #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2)$]
- To square a binomial: square the first term, square the last term, double their product. #linebreak()  #linebreak()
- #strong[Product of Conjugates Pattern]
- If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, #linebreak() 
#figure(figph[The image illustrates product of conjugates pattern with conjugates (a minus b) multiplied by (a plus b) equaling squares of a squared minus b squared.], alt: "The image illustrates product of conjugates pattern with conjugates (a minus b) multiplied by (a plus b) equaling squares of a squared minus b squared.", caption: none)
- #math.equation(block: false, 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. #linebreak()  #linebreak()
- #strong[To multiply conjugates:]
- #strong[square the first term square the last term] write it as a difference of squares


==== Practice Makes Perfect

#strong[Square a Binomial Using the Binomial Squares Pattern]

In the following exercises, square each binomial using the Binomial Squares Pattern.

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

#math.equation(block: true, alt: "open parenthesis q plus 12 close parenthesis squared")[$attach(( q + 12 ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "q squared plus 24 q plus 144")[$q^(2) + 24 q + 144$]
]

#math.equation(block: true, alt: "open parenthesis y plus the fraction 1 over 4 close parenthesis squared")[$attach(( y + frac(1, 4) ), t: 2)$]

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

#solutionbox[
#math.equation(block: true, alt: "x squared plus the fraction 4 over 3 x plus the fraction 4 over 9")[$x^(2) + frac(4, 3) x + frac(4, 9)$]
]

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

#math.equation(block: true, alt: "open parenthesis y minus 6 close parenthesis squared")[$attach(( y − 6 ), t: 2)$]

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

#math.equation(block: true, alt: "open parenthesis m minus 15 close parenthesis squared")[$attach(( m − 15 ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis p minus 13 close parenthesis squared")[$attach(( p − 13 ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "p squared minus 26 p plus 169")[$p^(2) − 26 p + 169$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "16 a squared plus 80 a plus 100")[$16 a^(2) + 80 a + 100$]
]

#math.equation(block: true, alt: "open parenthesis 2 q plus the fraction 1 over 3 close parenthesis squared")[$attach(( 2 q + frac(1, 3) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 3 z plus the fraction 1 over 5 close parenthesis squared")[$attach(( 3 z + frac(1, 5) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "9 z squared plus the fraction 6 over 5 z plus the fraction 1 over 25")[$9 z^(2) + frac(6, 5) z + frac(1, 25)$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "4 y squared minus 12 y z plus 9 z squared")[$4 y^(2) − 12 y z + 9 z^(2)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 5 x minus the fraction 1 over 7 y close parenthesis squared")[$attach(( frac(1, 5) x − frac(1, 7) y ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 8 x minus the fraction 1 over 9 y close parenthesis squared")[$attach(( frac(1, 8) x − frac(1, 9) y ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 64 x squared minus the fraction 1 over 36 x y plus the fraction 1 over 81 y squared")[$frac(1, 64) x^(2) − frac(1, 36) x y + frac(1, 81) y^(2)$]
]

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

#math.equation(block: true, alt: "open parenthesis 5 u squared plus 9 close parenthesis squared")[$attach(( 5 u^(2) + 9 ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "25 u to the power 4 plus 90 u squared plus 81")[$25 u^(4) + 90 u^(2) + 81$]
]

#math.equation(block: true, alt: "open parenthesis 4 y cubed minus 2 close parenthesis squared")[$attach(( 4 y^(3) − 2 ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 8 p cubed minus 3 close parenthesis squared")[$attach(( 8 p^(3) − 3 ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "64 p to the power 6 minus 48 p cubed plus 9")[$64 p^(6) − 48 p^(3) + 9$]
]

#strong[Multiply Conjugates Using the Product of Conjugates Pattern]

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

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

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

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

#math.equation(block: true, alt: "open parenthesis x plus the fraction 3 over 4 close parenthesis open parenthesis x minus the fraction 3 over 4 close parenthesis")[$( x + frac(3, 4) ) ( x − frac(3, 4) )$]

#math.equation(block: true, alt: "open parenthesis b plus the fraction 6 over 7 close parenthesis open parenthesis b minus the fraction 6 over 7 close parenthesis")[$( b + frac(6, 7) ) ( b − frac(6, 7) )$]

#solutionbox[
#math.equation(block: true, alt: "b squared minus the fraction 36 over 49")[$b^(2) − frac(36, 49)$]
]

#math.equation(block: true, alt: "open parenthesis 5 k plus 6 close parenthesis open parenthesis 5 k minus 6 close parenthesis")[$( 5 k + 6 ) ( 5 k − 6 )$]

#math.equation(block: true, alt: "open parenthesis 8 j plus 4 close parenthesis open parenthesis 8 j minus 4 close parenthesis")[$( 8 j + 4 ) ( 8 j − 4 )$]

#solutionbox[
#math.equation(block: true, alt: "64 j squared minus 16")[$64 j^(2) − 16$]
]

#math.equation(block: true, alt: "open parenthesis 11 k plus 4 close parenthesis open parenthesis 11 k minus 4 close parenthesis")[$( 11 k + 4 ) ( 11 k − 4 )$]

#math.equation(block: true, alt: "open parenthesis 9 c plus 5 close parenthesis open parenthesis 9 c minus 5 close parenthesis")[$( 9 c + 5 ) ( 9 c − 5 )$]

#solutionbox[
#math.equation(block: true, alt: "81 c squared minus 25")[$81 c^(2) − 25$]
]

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "16 minus 36 y squared")[$16 − 36 y^(2)$]
]

#math.equation(block: true, alt: "open parenthesis 9 c minus 2 d close parenthesis open parenthesis 9 c plus 2 d close parenthesis")[$( 9 c − 2 d ) ( 9 c + 2 d )$]

#math.equation(block: true, alt: "open parenthesis 7 w plus 10 x close parenthesis open parenthesis 7 w minus 10 x close parenthesis")[$( 7 w + 10 x ) ( 7 w − 10 x )$]

#solutionbox[
#math.equation(block: true, alt: "49 w squared minus 100 x squared")[$49 w^(2) − 100 x^(2)$]
]

#math.equation(block: true, alt: "open parenthesis m plus the fraction 2 over 3 n close parenthesis open parenthesis m minus the fraction 2 over 3 n close parenthesis")[$( m + frac(2, 3) n ) ( m − frac(2, 3) n )$]

#math.equation(block: true, alt: "open parenthesis p plus the fraction 4 over 5 q close parenthesis open parenthesis p minus the fraction 4 over 5 q close parenthesis")[$( p + frac(4, 5) q ) ( p − frac(4, 5) q )$]

#solutionbox[
#math.equation(block: true, alt: "p squared minus the fraction 16 over 25 q squared")[$p^(2) − frac(16, 25) q^(2)$]
]

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

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

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

#math.equation(block: true, alt: "open parenthesis u v minus the fraction 3 over 5 close parenthesis open parenthesis u v plus the fraction 3 over 5 close parenthesis")[$( u v − frac(3, 5) ) ( u v + frac(3, 5) )$]

#math.equation(block: true, alt: "open parenthesis r s minus the fraction 2 over 7 close parenthesis open parenthesis r s plus the fraction 2 over 7 close parenthesis")[$( r s − frac(2, 7) ) ( r s + frac(2, 7) )$]

#solutionbox[
#math.equation(block: true, alt: "r squared s squared minus the fraction 4 over 49")[$r^(2) s^(2) − frac(4, 49)$]
]

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

#math.equation(block: true, alt: "open parenthesis 6 m cubed minus 4 n to the power 5 close parenthesis open parenthesis 6 m cubed plus 4 n to the power 5 close parenthesis")[$( 6 m^(3) − 4 n^(5) ) ( 6 m^(3) + 4 n^(5) )$]

#solutionbox[
#math.equation(block: true, alt: "36 m to the power 6 minus 16 n to the power 10")[$36 m^(6) − 16 n^(10)$]
]

#math.equation(block: true, alt: "open parenthesis 12 p cubed minus 11 q squared close parenthesis open parenthesis 12 p cubed plus 11 q squared close parenthesis")[$( 12 p^(3) − 11 q^(2) ) ( 12 p^(3) + 11 q^(2) )$]

#math.equation(block: true, alt: "open parenthesis 15 m squared minus 8 n to the power 4 close parenthesis open parenthesis 15 m squared plus 8 n to the power 4 close parenthesis")[$( 15 m^(2) − 8 n^(4) ) ( 15 m^(2) + 8 n^(4) )$]

#solutionbox[
#math.equation(block: true, alt: "225 m to the power 4 minus 64 n to the power 8")[$225 m^(4) − 64 n^(8)$]
]

#strong[Recognize and Use the Appropriate Special Product Pattern]

In the following exercises, find each product.

ⓐ #math.equation(block: false, alt: "open parenthesis p minus 3 close parenthesis open parenthesis p plus 3 close parenthesis")[$( p − 3 ) ( p + 3 )$]
ⓑ #math.equation(block: false, alt: "open parenthesis t minus 9 close parenthesis squared")[$attach(( t − 9 ), t: 2)$]
ⓒ #math.equation(block: false, alt: "open parenthesis m plus n close parenthesis squared")[$attach(( m + n ), t: 2)$]
ⓓ #math.equation(block: false, alt: "open parenthesis 2 x plus y close parenthesis open parenthesis x minus 2 y close parenthesis")[$( 2 x + y ) ( x − 2 y )$]

ⓐ #math.equation(block: false, alt: "open parenthesis 2 r plus 12 close parenthesis squared")[$attach(( 2 r + 12 ), t: 2)$]

ⓑ #math.equation(block: false, alt: "open parenthesis 3 p plus 8 close parenthesis open parenthesis 3 p minus 8 close parenthesis")[$( 3 p + 8 ) ( 3 p − 8 )$]

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

 ⓓ #math.equation(block: false, alt: "open parenthesis k minus 6 close parenthesis squared")[$attach(( k − 6 ), t: 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4 r squared plus 48 r plus 144")[$4 r^(2) + 48 r + 144$] ⓑ #math.equation(block: false, alt: "9 p squared minus 64")[$9 p^(2) − 64$] ⓒ #math.equation(block: false, alt: "7 a squared minus 48 a b minus 7 b squared")[$7 a^(2) − 48 a b − 7 b^(2)$] ⓓ #math.equation(block: false, alt: "k squared minus 12 k plus 36")[$k^(2) − 12 k + 36$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis a to the power 5 minus 7 b close parenthesis squared")[$attach(( a^(5) − 7 b ), t: 2)$]

ⓑ #math.equation(block: false, alt: "open parenthesis x squared plus 8 y close parenthesis open parenthesis 8 x minus y squared close parenthesis")[$( x^(2) + 8 y ) ( 8 x − y^(2) )$]

ⓒ #math.equation(block: false, alt: "open parenthesis r to the power 6 plus s to the power 6 close parenthesis open parenthesis r to the power 6 minus s to the power 6 close parenthesis")[$( r^(6) + s^(6) ) ( r^(6) − s^(6) )$]

ⓓ #math.equation(block: false, alt: "open parenthesis y to the power 4 plus 2 z close parenthesis squared")[$attach(( y^(4) + 2 z ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 5 plus y to the power 5 close parenthesis open parenthesis x to the power 5 minus y to the power 5 close parenthesis")[$( x^(5) + y^(5) ) ( x^(5) − y^(5) )$]

ⓑ #math.equation(block: false, alt: "open parenthesis m cubed minus 8 n close parenthesis squared")[$attach(( m^(3) − 8 n ), t: 2)$]

ⓒ #math.equation(block: false, alt: "open parenthesis 9 p plus 8 q close parenthesis squared")[$attach(( 9 p + 8 q ), t: 2)$]

ⓓ #math.equation(block: false, alt: "open parenthesis r squared minus s cubed close parenthesis open parenthesis r cubed plus s squared close parenthesis")[$( r^(2) − s^(3) ) ( r^(3) + s^(2) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "x to the power 10 minus y to the power 10")[$x^(10) − y^(10)$] ⓑ #math.equation(block: false, alt: "m to the power 6 minus 16 m cubed n plus 64 n squared")[$m^(6) − 16 m^(3) n + 64 n^(2)$] ⓒ #math.equation(block: false, alt: "81 p squared plus 144 p q plus 64 q squared")[$81 p^(2) + 144 p q + 64 q^(2)$] ⓓ #math.equation(block: false, alt: "r to the power 5 plus r squared s squared minus r cubed s cubed minus s to the power 5")[$r^(5) + r^(2) s^(2) − r^(3) s^(3) − s^(5)$]
]


==== Everyday Math

#strong[Mental math] You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as #math.equation(block: false, alt: "50 minus 3")[$50 − 3$] and 53 as #math.equation(block: false, alt: "50 plus 3")[$50 + 3$].

+ ⓐ Multiply #math.equation(block: false, alt: "open parenthesis 50 minus 3 close parenthesis open parenthesis 50 plus 3 close parenthesis")[$( 50 − 3 ) ( 50 + 3 )$] by using the product of conjugates pattern, #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared")[$( a − b ) ( a + b ) = a^(2) − b^(2)$].
+ ⓑ Multiply #math.equation(block: false, alt: "47 times 53")[$47 · 53$] without using a calculator.
+ ⓒ Which way is easier for you? Why?

#strong[Mental math] You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as #math.equation(block: false, alt: "60 plus 5")[$60 + 5$].

+ ⓐ Multiply #math.equation(block: false, alt: "open parenthesis 60 plus 5 close parenthesis squared")[$attach(( 60 + 5 ), t: 2)$] by using the binomial squares pattern, #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2)$].
+ ⓑ Square 65 without using a calculator.
+ ⓒ Which way is easier for you? Why?

#solutionbox[
ⓐ 4,225 ⓑ 4,225 ⓒ Answers will vary.
]


==== Writing Exercises

How do you decide which pattern to use?

Why does #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared")[$attach(( a + b ), t: 2)$] result in a trinomial, but #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis")[$( a − b ) ( a + b )$] result in a binomial?

#solutionbox[
Answers will vary.
]

Marta did the following work on her homework paper:

#math.equation(block: true, alt: "open parenthesis 3 minus y close parenthesis squared; 3 squared minus y squared; 9 minus y squared")[$attach(( 3 − y ), t: 2) \ #h(0.2em) 3^(2) − y^(2) \ #h(0.7em) 9 − y^(2)$]Explain what is wrong with Marta’s work.

Use the order of operations to show that #math.equation(block: false, alt: "open parenthesis 3 plus 5 close parenthesis squared")[$attach(( 3 + 5 ), t: 2)$] is 64, and then use that numerical example to explain why #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared not equal to a squared plus b squared")[$attach(( a + b ), t: 2) ≠ a^(2) + b^(2)$].

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “square a binomial using the binomial squares pattern,” “multiply conjugates using the product of conjugates pattern,” and “recognize and use the appropriate special product pattern.” The rest of the cells are blank.], alt: "This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “square a binomial using the binomial squares pattern,” “multiply conjugates using the product of conjugates pattern,” and “recognize and use the appropriate special product pattern.” The rest of the cells are blank.", 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?