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== 10.3#h(0.6em)Multiply Polynomials

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

Distribute: #math.equation(block: false, alt: "2 open parenthesis x plus 3 close parenthesis .")[$2 ( x + 3 ) .$] #linebreak() 
If you missed the problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 x plus 6")[$2 x + 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Distribute: #math.equation(block: false, alt: "−11 open parenthesis 4 minus 3 a close parenthesis .")[$−11 ( 4 − 3 a ) .$] #linebreak() 
If you missed the problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 44 plus 33 a")[$− 44 + 33 a$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Combine like terms: #math.equation(block: false, alt: "x squared plus 9 x plus 7 x plus 63 .")[$x^(2) + 9 x + 7 x + 63 .$] #linebreak() 
If you missed the problem, review .

#solutionbox[
#math.equation(block: true, alt: "x squared plus 16 x plus 63")[$x^(2) + 16 x + 63$]
]
]


=== Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the #strong[Distributive Property] to simplify expressions such as #math.equation(block: false, alt: "2 open parenthesis x minus 3 close parenthesis .")[$2 ( x − 3 ) .$] You multiplied both terms in the parentheses, #math.equation(block: false, alt: "x and 3 ,")[$x #h(0.2em) "and" #h(0.2em) 3 ,$] by #math.equation(block: false, alt: "2 ,")[$2 ,$] to get #math.equation(block: false, alt: "2 x minus 6 .")[$2 x − 6 .$] With this chapter's new vocabulary, you can say you were multiplying a binomial, #math.equation(block: false, alt: "x minus 3 ,")[$x − 3 ,$] by a monomial, #math.equation(block: false, alt: "2 .")[$2 .$] Multiplying a binomial by a monomial is nothing new for you!

#examplebox("Example 1")[][
Multiply: #math.equation(block: false, alt: "3 open parenthesis x plus 7 close parenthesis .")[$3 ( x + 7 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 open parenthesis x plus 7 close parenthesis")[$3 ( x + 7 )$]]),
  [Distribute.], [#figure(figph[An image illustrating the distributive property in algebra, showing red arrows indicating how to multiply 3 by both 'x' and '7' in the expression 3(x + 7).], alt: "An image illustrating the distributive property in algebra, showing red arrows indicating how to multiply 3 by both 'x' and '7' in the expression 3(x + 7).", caption: none)],
  [], [#math.equation(block: false, alt: "3 times x plus 3 times 7")[$3 · x + 3 · 7$]],
  [Simplify.], [#math.equation(block: false, alt: "3 x plus 21")[$3 x + 21$]],
))
]
]

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

#solutionbox[
6#emph[x] + 48
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "2 open parenthesis y plus 12 close parenthesis .")[$2 ( y + 12 ) .$]

#solutionbox[
2#emph[y] + 24
]
]

#examplebox("Example 2")[][
Multiply: #math.equation(block: false, alt: "x open parenthesis x minus 8 close parenthesis .")[$x ( x − 8 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression reads x(x-8) in black text on a white background.], alt: "A mathematical expression reads x(x-8) in black text on a white background.", caption: none)]),
  [Distribute.], [#figure(figph[Red arrows indicate the distributive property for the expression x(x - 8), showing that the outer 'x' multiplies each term within the parentheses.], alt: "Red arrows indicate the distributive property for the expression x(x - 8), showing that the outer 'x' multiplies each term within the parentheses.", caption: none)],
  [], [#figure(figph[The mathematical expression x squared minus 8x, written in black font on a white background.], alt: "The mathematical expression x squared minus 8x, written in black font on a white background.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression x squared minus 8x, written in black font on a white background.], alt: "The mathematical expression x squared minus 8x, written in black font on a white background.", caption: none)],
))
]
]

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

#solutionbox[
#emph[y]#super[2] − 9#emph[y]
]
]

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

#solutionbox[
#emph[p]#super[2] − 13#emph[p]
]
]

#examplebox("Example 3")[][
Multiply: #math.equation(block: false, alt: "10 x open parenthesis 4 x plus y close parenthesis .")[$10 x ( 4 x + y ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the algebraic expression 10x(4x + y).], alt: "The image shows the algebraic expression 10x(4x + y).", caption: none)]),
  [Distribute.], [#figure(figph[An algebraic expression 10x(4x + y) is shown with red curved arrows illustrating the distributive property, pointing from 10x to both 4x and y inside the parentheses.], alt: "An algebraic expression 10x(4x + y) is shown with red curved arrows illustrating the distributive property, pointing from 10x to both 4x and y inside the parentheses.", caption: none)],
  [], [#figure(figph[The image shows the mathematical expression 10x times 4x plus 10x times y.], alt: "The image shows the mathematical expression 10x times 4x plus 10x times y.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical expression 40x^2 + 10xy, rendered in a black serif font against a plain white background.], alt: "The image shows the mathematical expression 40x^2 + 10xy, rendered in a black serif font against a plain white background.", caption: none)],
))
]
]

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

#solutionbox[
8#emph[x]#super[2] + 24#emph[xy]
]
]

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

#solutionbox[
18#emph[r]#super[2] + 3#emph[rs]
]
]

Multiplying a monomial by a #strong[trinomial] works in much the same way.

#examplebox("Example 4")[][
Multiply: #math.equation(block: false, alt: "−2 x open parenthesis 5 x squared plus 7 x minus 3 close parenthesis .")[$−2 x ( 5 x^(2) + 7 x − 3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−2 x open parenthesis 5 x squared plus 7 x minus 3 close parenthesis")[$−2 x ( 5 x^(2) + 7 x − 3 )$]]),
  [Distribute.], [#figure(figph[The image shows the distributive property being applied to the expression -2x(5x^2 + 7x - 3), with red arrows illustrating how -2x is multiplied by each term inside the parentheses.], alt: "The image shows the distributive property being applied to the expression -2x(5x^2 + 7x - 3), with red arrows illustrating how -2x is multiplied by each term inside the parentheses.", caption: none)],
  [], [#math.equation(block: false, alt: "−2 x times 5 x squared plus open parenthesis −2 x close parenthesis times 7 x minus open parenthesis −2 x close parenthesis times 3")[$−2 x ⋅ 5 x^(2) + ( −2 x ) ⋅ 7 x − ( −2 x ) ⋅ 3$]],
  [Simplify.], [#math.equation(block: false, alt: "−10 x cubed minus 14 x squared plus 6 x")[$−10 x^(3) − 14 x^(2) + 6 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "−4 y open parenthesis 8 y squared plus 5 y minus 9 close parenthesis .")[$−4 y ( 8 y^(2) + 5 y − 9 ) .$]

#solutionbox[
−32#emph[y]#super[3] − 20#emph[y]#super[2] + 36#emph[y]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "−6 x open parenthesis 9 x squared plus x minus 1 close parenthesis .")[$−6 x ( 9 x^(2) + x − 1 ) .$]

#solutionbox[
−54#emph[x]#super[3] − 6#emph[x]#super[2] + 6#emph[x]
]
]

#examplebox("Example 5")[][
Multiply: #math.equation(block: false, alt: "4 y cubed open parenthesis y squared minus 8 y plus 1 close parenthesis .")[$4 y^(3) ( y^(2) − 8 y + 1 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 y cubed open parenthesis y squared minus 8 y plus 1 close parenthesis")[$4 y^(3) ( y^(2) − 8 y + 1 )$]]),
  [Distribute.], [#figure(figph[An algebraic expression shows the distributive property. The term 4y^3 is multiplied by each term inside the parentheses (y^2 - 8y + 1), indicated by red curved arrows.], alt: "An algebraic expression shows the distributive property. The term 4y^3 is multiplied by each term inside the parentheses (y^2 - 8y + 1), indicated by red curved arrows.", caption: none)],
  [], [#math.equation(block: false, alt: "4 y cubed times y squared minus 4 y cubed times 8 y plus 4 y cubed times 1")[$4 y^(3) ⋅ y^(2) − 4 y^(3) ⋅ 8 y + 4 y^(3) ⋅ 1$]],
  [Simplify.], [#math.equation(block: false, alt: "4 y to the power 5 minus 32 y to the power 4 plus 4 y cubed")[$4 y^(5) − 32 y^(4) + 4 y^(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "3 x squared open parenthesis 4 x squared minus 3 x plus 9 close parenthesis .")[$3 x^(2) ( 4 x^(2) − 3 x + 9 ) .$]

#solutionbox[
12#emph[x]#super[4] − 9#emph[x]#super[3] + 27#emph[x]#super[2]
]
]

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

#solutionbox[
24#emph[y]#super[4] − 16#emph[y]#super[3] − 32#emph[y]#super[2]
]
]

Now we will have the #strong[monomial] as the second factor.

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis p")[$( x + 3 ) p$]]),
  [Distribute.], [#figure(figph[Illustrating the distributive property in algebra, showing p being multiplied by x and by 3 in the expression (x+3)p.], alt: "Illustrating the distributive property in algebra, showing p being multiplied by x and by 3 in the expression (x+3)p.", caption: none)],
  [], [#math.equation(block: false, alt: "x times p plus 3 times p")[$x ⋅ p + 3 ⋅ p$]],
  [Simplify.], [#math.equation(block: false, alt: "x p plus 3 p")[$x p + 3 p$]],
))
]
]

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

#solutionbox[
#emph[xp] + 8#emph[p]
]
]

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

#solutionbox[
#emph[ap] + 4#emph[p]
]
]


=== Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.


==== Using the Distributive Property

We will start by using the #strong[Distributive Property]. Look again at .

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic expression (x+3)p, demonstrating the distributive property with curved red arrows indicating that 'p' multiplies both 'x' and '3' within the parentheses.], alt: "An algebraic expression (x+3)p, demonstrating the distributive property with curved red arrows indicating that 'p' multiplies both 'x' and '3' within the parentheses.", caption: none)]),
  [We distributed the #math.equation(block: false, alt: "p")[$p$] to get], [#figure(figph[The mathematical expression xp + 3p is shown, featuring the variables 'x' and 'p' with 'p' highlighted in red, indicating a common factor.], alt: "The mathematical expression xp + 3p is shown, featuring the variables 'x' and 'p' with 'p' highlighted in red, indicating a common factor.", caption: none)],
  [What if we have #math.equation(block: false, alt: "open parenthesis x plus 7 close parenthesis")[$( x + 7 )$] instead of #math.equation(block: false, alt: "p")[$p$]?
    #linebreak() #figure(figph[The image displays the text 'Think of the (x + 7) as the p above.' in a teal-like font, with the letter 'p' highlighted in red.], alt: "The image displays the text 'Think of the (x + 7) as the p above.' in a teal-like font, with the letter 'p' highlighted in red.", caption: none)], [#figure(figph[The distributive property (FOIL method) applied to the binomial multiplication (x+3)(x+7), with red arrows showing the term distribution.], alt: "The distributive property (FOIL method) applied to the binomial multiplication (x+3)(x+7), with red arrows showing the term distribution.", caption: none)],
  [Distribute #math.equation(block: false, alt: "open parenthesis x plus 7 close parenthesis")[$( x + 7 )$].], [#figure(figph[An algebraic expression showing x(x + 7) + 3(x + 7) with red arrows illustrating the distributive property applied to both terms.], alt: "An algebraic expression showing x(x + 7) + 3(x + 7) with red arrows illustrating the distributive property applied to both terms.", caption: none)],
  [Distribute again.], [#math.equation(block: false, alt: "x squared plus 7 x plus 3 x plus 21")[$x^(2) + 7 x + 3 x + 21$]],
  [Combine like terms.], [#math.equation(block: false, alt: "x squared plus 10 x plus 21")[$x^(2) + 10 x + 21$]],
))

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

Be careful to distinguish between a sum and a product.

#math.equation(block: true, alt: "Sum, Product; x plus x, x times x; 2 x, x squared; combine like terms, add exponents of like bases")[$"Sum" &  &  & "Product" \ x + x &  &  & x · x \ 2 x &  &  & x^(2) \ "combine like terms" &  &  & "add exponents of like bases"$]#examplebox("Example 7")[][
Multiply: #math.equation(block: false, alt: "open parenthesis x plus 6 close parenthesis open parenthesis x plus 8 close parenthesis .")[$( x + 6 ) ( x + 8 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis x plus 6 close parenthesis open parenthesis x plus 8 close parenthesis")[$( x + 6 ) ( x + 8 )$]]),
  [], [#figure(figph[This image shows the initial steps of multiplying two binomials, (x+6)(x+8), using the distributive property. Red arrows demonstrate 'x' from the first term multiplying 'x' and '8' in the second term.], alt: "This image shows the initial steps of multiplying two binomials, (x+6)(x+8), using the distributive property. Red arrows demonstrate 'x' from the first term multiplying 'x' and '8' in the second term.", caption: none)],
  [Distribute #math.equation(block: false, alt: "open parenthesis x plus 8 close parenthesis")[$( x + 8 )$].], [#figure(figph[A mathematical expression shows x multiplied by the quantity (x plus 8), added to 6 multiplied by the quantity (x plus 8).], alt: "A mathematical expression shows x multiplied by the quantity (x plus 8), added to 6 multiplied by the quantity (x plus 8).", caption: none)],
  [Distribute again.], [#math.equation(block: false, alt: "x squared plus 8 x plus 6 x plus 48")[$x^(2) + 8 x + 6 x + 48$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared plus 14 x plus 48")[$x^(2) + 14 x + 48$]],
))
]
]

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

#solutionbox[
#emph[x]#super[2] + 17#emph[x] + 72
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis a plus 4 close parenthesis open parenthesis a plus 5 close parenthesis .")[$( a + 4 ) ( a + 5 ) .$]

#solutionbox[
#emph[a]#super[2] + 9#emph[a] + 20
]
]

Now we'll see how to multiply binomials where the variable has a #strong[coefficient].

#examplebox("Example 8")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 2 x plus 9 close parenthesis open parenthesis 3 x plus 4 close parenthesis .")[$( 2 x + 9 ) ( 3 x + 4 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 2 x plus 9 close parenthesis open parenthesis 3 x plus 4 close parenthesis")[$( 2 x + 9 ) ( 3 x + 4 )$]]),
  [Distribute. #math.equation(block: false, alt: "open parenthesis 3 x plus 4 close parenthesis")[$( 3 x + 4 )$]], [#figure(figph[A mathematical expression shows 2x multiplied by the quantity 3x plus 4, added to 9 multiplied by the quantity 3x plus 4. The terms (3x + 4) are highlighted in red, indicating a common factor.], alt: "A mathematical expression shows 2x multiplied by the quantity 3x plus 4, added to 9 multiplied by the quantity 3x plus 4. The terms (3x + 4) are highlighted in red, indicating a common factor.", caption: none)],
  [Distribute again.], [#math.equation(block: false, alt: "6 x squared plus 8 x plus 27 x plus 36")[$6 x^(2) + 8 x + 27 x + 36$]],
  [Simplify.], [#math.equation(block: false, alt: "6 x squared plus 35 x plus 36")[$6 x^(2) + 35 x + 36$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 5 x plus 9 close parenthesis open parenthesis 4 x plus 3 close parenthesis .")[$( 5 x + 9 ) ( 4 x + 3 ) .$]

#solutionbox[
20#emph[x]#super[2] + 51#emph[x] + 27
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 10 m plus 9 close parenthesis open parenthesis 8 m plus 7 close parenthesis .")[$( 10 m + 9 ) ( 8 m + 7 ) .$]

#solutionbox[
80#emph[m]#super[2] + 142#emph[m] + 63
]
]

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

#examplebox("Example 9")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 4 y plus 3 close parenthesis open parenthesis 6 y minus 5 close parenthesis .")[$( 4 y + 3 ) ( 6 y − 5 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 4 y plus 3 close parenthesis open parenthesis 6 y minus 5 close parenthesis")[$( 4 y + 3 ) ( 6 y − 5 )$]]),
  [Distribute.], [#figure(figph[The image shows the mathematical expression 4y(6y - 5) + 3(6y - 5), which can be simplified by factoring out the common term (6y - 5).], alt: "The image shows the mathematical expression 4y(6y - 5) + 3(6y - 5), which can be simplified by factoring out the common term (6y - 5).", caption: none)],
  [Distribute again.], [#math.equation(block: false, alt: "24 y squared minus 20 y plus 18 y minus 15")[$24 y^(2) − 20 y + 18 y − 15$]],
  [Simplify.], [#math.equation(block: false, alt: "24 y squared minus 2 y minus 15")[$24 y^(2) − 2 y − 15$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 7 y plus 1 close parenthesis open parenthesis 8 y minus 3 close parenthesis .")[$( 7 y + 1 ) ( 8 y − 3 ) .$]

#solutionbox[
56#emph[y]#super[2] − 13#emph[y] − 3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 3 x plus 2 close parenthesis open parenthesis 5 x minus 8 close parenthesis .")[$( 3 x + 2 ) ( 5 x − 8 ) .$]

#solutionbox[
15#emph[x]#super[2] − 14#emph[x] − 16
]
]

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

#examplebox("Example 10")[][
Multiply: #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus y close parenthesis .")[$( x + 2 ) ( x − y ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression in black text on a white background, showing the product of two binomials: (x + 2)(x - y).], alt: "A mathematical expression in black text on a white background, showing the product of two binomials: (x + 2)(x - y).", caption: none)]),
  [Distribute.], [#figure(figph[The image displays the mathematical expression x(x - y) + 2(x - y), which involves variables x and y, parentheses, multiplication, addition, and subtraction.], alt: "The image displays the mathematical expression x(x - y) + 2(x - y), which involves variables x and y, parentheses, multiplication, addition, and subtraction.", caption: none)],
  [Distribute again.], [#figure(figph[The image displays the algebraic expression x^2 - xy + 2x - 2y, which is a polynomial with four terms involving variables x and y, and constant coefficients.], alt: "The image displays the algebraic expression x^2 - xy + 2x - 2y, which is a polynomial with four terms involving variables x and y, and constant coefficients.", caption: none)],
  [Simplify.], [There are no like terms to combine.],
))
]
]

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

#solutionbox[
#emph[x]#super[2] − #emph[xy] + 5#emph[x] − 5#emph[y]
]
]

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

#solutionbox[
#emph[x]#super[2] − #emph[x] + 2#emph[xy] − 2#emph[y]
]
]


==== Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.

#math.equation(block: true, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x minus y close parenthesis")[$( x + 2 ) ( x − y )$]#math.equation(block: true, alt: "x squared minus xy plus 2 x minus 2 y")[$x^(2) − "xy" + 2 x − 2 y$]Where did the first term, #math.equation(block: false, alt: "x squared ,")[$x^(2) ,$] come from?

It is the product of #math.equation(block: false, alt: "x and x ,")[$x #h(0.2em) "and" #h(0.2em) x ,$] the #strong[first] terms in #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis and open parenthesis x minus y close parenthesis .")[$( x + 2 ) #h(0.2em) "and" #h(0.2em) ( x − y ) .$]

#figure(figph[Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this, “First” is written in red.], alt: "Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this, “First” is written in red.", caption: none)

The next term, #math.equation(block: false, alt: "minus xy ,")[$− "xy" ,$] is the product of #math.equation(block: false, alt: "x and minus y ,")[$x #h(0.2em) "and" #h(0.2em) − y ,$] the two #strong[outer] terms.

#figure(figph[Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this, “Outer” is written in red.], alt: "Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this, “Outer” is written in red.", caption: none)

The third term, #math.equation(block: false, alt: "+2 x ,")[$+2 x ,$] is the product of #math.equation(block: false, alt: "2 and x ,")[$2 #h(0.2em) "and" #h(0.2em) x ,$] the two #strong[inner] terms.

#figure(figph[Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that, “Inner” is written in red.], alt: "Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that, “Inner” is written in red.", caption: none)

And the last term, #math.equation(block: false, alt: "−2 y ,")[$−2 y ,$] came from multiplying the two #strong[last] terms.

#figure(figph[Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that, “Last” is written in red.], alt: "Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that, “Last” is written in red.", caption: none)

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.

#figure(figph[Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.], alt: "Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.", caption: none)

Let's look at #math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x plus 7 close parenthesis")[$( x + 3 ) ( x + 7 )$] again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

#figure(figph[Comparing the Distributive Property and FOIL method for multiplying binomials, like (x+3)(x+7), both resulting in x^2 + 10x + 21.], alt: "Comparing the Distributive Property and FOIL method for multiplying binomials, like (x+3)(x+7), both resulting in x^2 + 10x + 21.", caption: none)

#examplebox("Example 11")[][
Multiply using the FOIL method: #math.equation(block: false, alt: "open parenthesis x plus 6 close parenthesis open parenthesis x plus 9 close parenthesis .")[$( x + 6 ) ( x + 9 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1]: Multiply the #strong[First] terms.], [#figure(figph[An algebraic expression (x + 6)(x + 9) being expanded using the FOIL method, with the first term x² explicitly shown and placeholders for the Outer, Inner, and Last terms.], alt: "An algebraic expression (x + 6)(x + 9) being expanded using the FOIL method, with the first term x² explicitly shown and placeholders for the Outer, Inner, and Last terms.", caption: none)]),
  [#strong[Step 2]: Multiply the #strong[Outer] terms.], [#figure(figph[This image demonstrates the 'First' step of the FOIL method for multiplying binomials (x+6)(x+9). The red arrow points from the 'x' in the first term to the 'x' in the second, showing x \* x = x^2. The partial result includes x^2 and the 'Outer' term, 9x, with spaces for the 'Inner' and 'Last' terms.], alt: "This image demonstrates the 'First' step of the FOIL method for multiplying binomials (x+6)(x+9). The red arrow points from the 'x' in the first term to the 'x' in the second, showing x * x = x^2. The partial result includes x^2 and the 'Outer' term, 9x, with spaces for the 'Inner' and 'Last' terms.", caption: none)],
  [#strong[Step 3]: Multiply the #strong[Inner] terms.], [#figure(figph[This image illustrates the FOIL method for multiplying binomials. It shows (x+6)(x+9) expanding to x^2 + 9x + 6x + \_\_\_, with the red arrow emphasizing the 'Inner' term multiplication of 6 and x.], alt: "This image illustrates the FOIL method for multiplying binomials. It shows (x+6)(x+9) expanding to x^2 + 9x + 6x + ___, with the red arrow emphasizing the 'Inner' term multiplication of 6 and x.", caption: none)],
  [#strong[Step 4]: Multiply the #strong[Last] terms.], [#figure(figph[The FOIL method for multiplying binomials is shown with (x+6)(x+9) expanding to x^2 + 9x + 6x + 54, clearly marking the First, Outer, Inner, and Last products.], alt: "The FOIL method for multiplying binomials is shown with (x+6)(x+9) expanding to x^2 + 9x + 6x + 54, clearly marking the First, Outer, Inner, and Last products.", caption: none)],
  [#strong[Step 5]: Combine like terms, when possible.], [#figure(figph[The image displays the quadratic expression x^2 + 15x + 54.], alt: "The image displays the quadratic expression x^2 + 15x + 54.", caption: none)],
))
]
]

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

#solutionbox[
#emph[x]#super[2] + 15#emph[x] + 56
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the FOIL method: #math.equation(block: false, alt: "open parenthesis y plus 14 close parenthesis open parenthesis y plus 2 close parenthesis .")[$( y + 14 ) ( y + 2 ) .$]

#solutionbox[
#emph[y]#super[2] + 16#emph[y] + 28
]
]

We summarize the steps of the FOIL method below. The #strong[FOIL method] only applies to multiplying binomials, not other polynomials!

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use the FOIL method for multiplying two binomials.]

+ Multiply the #strong[First] terms.
+ Multiply the #strong[Outer] terms.
+ Multiply the #strong[Inner] terms.
+ Multiply the #strong[Last] terms.
+ Combine like terms, when possible.

#figure(figph[Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.], alt: "Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.", caption: none)
]

#examplebox("Example 12")[][
Multiply: #math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis open parenthesis y plus 6 close parenthesis .")[$( y − 8 ) ( y + 6 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1]: Multiply the #strong[First] terms.], [#figure(figph[This image demonstrates the FOIL method for multiplying binomials. It shows (y-8)(y+6) expanding to y^2 + \_ + \_ + \_ with F, O, I, L terms indicated below, where y^2 is the 'First' term.], alt: "This image demonstrates the FOIL method for multiplying binomials. It shows (y-8)(y+6) expanding to y^2 + _ + _ + _ with F, O, I, L terms indicated below, where y^2 is the 'First' term.", caption: none)]),
  [#strong[Step 2]: Multiply the #strong[Outer] terms.], [#figure(figph[The image illustrates the FOIL method for multiplying binomials (y-8)(y+6). Arrows highlight the First (y\*y), Outer (y\*6), Inner (-8\*y), and Last (-8\*6) terms. The partial expansion y^2 + 6y + \_ + \_ is labeled with F, O, I, L.], alt: "The image illustrates the FOIL method for multiplying binomials (y-8)(y+6). Arrows highlight the First (y*y), Outer (y*6), Inner (-8*y), and Last (-8*6) terms. The partial expansion y^2 + 6y + _ + _ is labeled with F, O, I, L.", caption: none)],
  [#strong[Step 3]: Multiply the #strong[Inner] terms.], [#figure(figph[An image illustrating the FOIL method for multiplying two binomials, (y - 8)(y + 6). It shows the 'First', 'Outer', and 'Inner' terms of the expansion y^2 + 6y - 8y, with a blank for the 'Last' term.], alt: "An image illustrating the FOIL method for multiplying two binomials, (y - 8)(y + 6). It shows the 'First', 'Outer', and 'Inner' terms of the expansion y^2 + 6y - 8y, with a blank for the 'Last' term.", caption: none)],
  [#strong[Step 4]: Multiply the #strong[Last] terms.], [#figure(figph[This image illustrates the FOIL method for multiplying two binomials: (y-8)(y+6). It shows how to expand it to y^2 + 6y - 8y - 48, indicating the 'First', 'Outer', 'Inner', and 'Last' terms with arrows and labels.], alt: "This image illustrates the FOIL method for multiplying two binomials: (y-8)(y+6). It shows how to expand it to y^2 + 6y - 8y - 48, indicating the 'First', 'Outer', 'Inner', and 'Last' terms with arrows and labels.", caption: none)],
  [#strong[Step 5]: Combine like terms], [#figure(figph[A mathematical expression, y squared minus 2y minus 48, is centered on a white background. It represents a quadratic polynomial.], alt: "A mathematical expression, y squared minus 2y minus 48, is centered on a white background. It represents a quadratic polynomial.", caption: none)],
))
]
]

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

#solutionbox[
#emph[y]#super[2] + 5#emph[y] − 24
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis q minus 4 close parenthesis open parenthesis q plus 5 close parenthesis .")[$( q − 4 ) ( q + 5 ) .$]

#solutionbox[
#emph[q]#super[2] + #emph[q] − 20
]
]

#examplebox("Example 13")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 2 a plus 3 close parenthesis open parenthesis 3 a minus 1 close parenthesis .")[$( 2 a + 3 ) ( 3 a − 1 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the mathematical expression (2a + 3)(3a - 1).], alt: "The image displays the mathematical expression (2a + 3)(3a - 1).", caption: none)]),
  [], [#figure(figph[The FOIL method is illustrated for multiplying the binomials (2a + 3) and (3a - 1), showing the distributive steps with curved arrows.], alt: "The FOIL method is illustrated for multiplying the binomials (2a + 3) and (3a - 1), showing the distributive steps with curved arrows.", caption: none)],
  [Multiply the #strong[First] terms.], [#figure(figph[A mathematics diagram illustrating the start of the FOIL method, showing the product of the first terms '2a \* 3a = 6a^2' with placeholders for Outer, Inner, and Last terms denoted by F, O, I, L.], alt: "A mathematics diagram illustrating the start of the FOIL method, showing the product of the first terms '2a * 3a = 6a^2' with placeholders for Outer, Inner, and Last terms denoted by F, O, I, L.", caption: none)],
  [Multiply the #strong[Outer] terms.], [#figure(figph[This image presents a math exercise on the FOIL method, displaying the First (6a^2) and Outer (-2a) terms of a binomial product. Blanks are provided for the Inner and Last terms, accompanied by a hint: 2a \* (-1).], alt: "This image presents a math exercise on the FOIL method, displaying the First (6a^2) and Outer (-2a) terms of a binomial product. Blanks are provided for the Inner and Last terms, accompanied by a hint: 2a * (-1).", caption: none)],
  [Multiply the #strong[Inner] terms.], [#figure(figph[Image showing an algebraic expression 6a^2 - 2a + 9a + \_\_ with F, O, I, L labels below, illustrating the FOIL method for multiplying binomials. A separate term '3 \* 3a' is also visible.], alt: "Image showing an algebraic expression 6a^2 - 2a + 9a + __ with F, O, I, L labels below, illustrating the FOIL method for multiplying binomials. A separate term '3 * 3a' is also visible.", caption: none)],
  [Multiply the #strong[Last] terms.], [#figure(figph[Mathematical notation showing the expansion of a binomial using the FOIL method, with terms 6a^2, -2a, +9a, and -3 labeled F, O, I, L respectively. A multiplication 3 \* (-1) is also visible.], alt: "Mathematical notation showing the expansion of a binomial using the FOIL method, with terms 6a^2, -2a, +9a, and -3 labeled F, O, I, L respectively. A multiplication 3 * (-1) is also visible.", caption: none)],
  [Combine like terms.], [#figure(figph[The image displays the quadratic expression 6a^2 + 7a - 3 in black text on a white background.], alt: "The image displays the quadratic expression 6a^2 + 7a - 3 in black text on a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 4 a plus 9 close parenthesis open parenthesis 5 a minus 2 close parenthesis .")[$( 4 a + 9 ) ( 5 a − 2 ) .$]

#solutionbox[
20#emph[a]#super[2] + 37#emph[a] − 18
]
]

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

#solutionbox[
49#emph[x]#super[2] − 28#emph[x] − 32
]
]

#examplebox("Example 14")[][
Multiply: #math.equation(block: false, alt: "open parenthesis 5 x minus y close parenthesis open parenthesis 2 x minus 7 close parenthesis .")[$( 5 x − y ) ( 2 x − 7 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the product of two binomials: (5x - y) and (2x - 7).], alt: "A mathematical expression showing the product of two binomials: (5x - y) and (2x - 7).", caption: none)]),
  [], [#figure(figph[An algebraic expression showing the multiplication of two binomials, (5x - y) and (2x - 7), with curved arrows illustrating the FOIL method for distribution.], alt: "An algebraic expression showing the multiplication of two binomials, (5x - y) and (2x - 7), with curved arrows illustrating the FOIL method for distribution.", caption: none)],
  [Multiply the #strong[First] terms.], [#figure(figph[An algebra example demonstrating the FOIL method for multiplying binomials, with '10x^2' as the 'F' (First) term and blanks for the 'O', 'I', and 'L' terms.], alt: "An algebra example demonstrating the FOIL method for multiplying binomials, with '10x^2' as the 'F' (First) term and blanks for the 'O', 'I', and 'L' terms.", caption: none)],
  [Multiply the #strong[Outer] terms.], [#figure(figph[An algebraic expression demonstrating the FOIL method, showing the 'First' term as 10x^2 and the 'Outer' term as -35x, with blanks representing the 'Inner' and 'Last' terms.], alt: "An algebraic expression demonstrating the FOIL method, showing the 'First' term as 10x^2 and the 'Outer' term as -35x, with blanks representing the 'Inner' and 'Last' terms.", caption: none)],
  [Multiply the #strong[Inner] terms.], [#figure(figph[A partial algebraic expression 10x^2 - 35x - 2xy + \_\_\_ with the letters F, O, I, L annotated below, illustrating the FOIL method where the 'L' (Last) term is missing.], alt: "A partial algebraic expression 10x^2 - 35x - 2xy + ___ with the letters F, O, I, L annotated below, illustrating the FOIL method where the 'L' (Last) term is missing.", caption: none)],
  [Multiply the #strong[Last] terms.], [#figure(figph[The polynomial expression 10x² - 35x - 2xy + 7y is displayed, with the letters F, O, I, L beneath its terms, referencing the FOIL method. The last term '+7y' is notably highlighted in red.], alt: "The polynomial expression 10x² - 35x - 2xy + 7y is displayed, with the letters F, O, I, L beneath its terms, referencing the FOIL method. The last term '+7y' is notably highlighted in red.", caption: none)],
  [Combine like terms. There are none.], [#figure(figph[A mathematical expression reads 10x^2 - 35x - 2xy + 7y. The numbers and variables are black except for the '+ 7y' at the end, which is rendered in red.], alt: "A mathematical expression reads 10x^2 - 35x - 2xy + 7y. The numbers and variables are black except for the '+ 7y' at the end, which is rendered in red.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 12 x minus y close parenthesis open parenthesis x minus 5 close parenthesis .")[$( 12 x − y ) ( x − 5 ) .$]

#solutionbox[
12#emph[x]#super[2] − 60#emph[x] − #emph[xy] + 5#emph[y]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 6 a minus b close parenthesis open parenthesis 2 a minus 9 close parenthesis .")[$( 6 a − b ) ( 2 a − 9 ) .$]

#solutionbox[
12#emph[a]#super[2] − 54#emph[a] − 2#emph[ab] + 9#emph[b]
]
]


==== Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works #emph[only] for binomials. You can use the #strong[Distributive Property] to find the product of any two polynomials. Another method that works for all polynomials is the #strong[Vertical Method]. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

#figure(figph[A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written “partial product.” Beneath 138 is 92. Beside 92 is written “partial product.” Beneath 92 is a line and 1058. Beside 1058 is written “product.”], alt: "A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written “partial product.” Beneath 138 is 92. Beside 92 is written “partial product.” Beneath 92 is a line and 1058. Beside 1058 is written “product.”", caption: none)

You start by multiplying #math.equation(block: false, alt: "23")[$23$] by #math.equation(block: false, alt: "6")[$6$] to get #math.equation(block: false, alt: "138 .")[$138 .$]

Then you multiply #math.equation(block: false, alt: "23")[$23$] by #math.equation(block: false, alt: "4 ,")[$4 ,$] lining up the partial product in the correct columns.

Last, you add the partial products.

Now we'll apply this same method to multiply two binomials.

#examplebox("Example 15")[][
Multiply using the vertical method: #math.equation(block: false, alt: "open parenthesis 5 x minus 1 close parenthesis open parenthesis 2 x minus 7 close parenthesis .")[$( 5 x − 1 ) ( 2 x − 7 ) .$]

#solutionbox[
It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied #math.equation(block: false, alt: "23 open parenthesis 46 close parenthesis .")[$23 ( 46 ) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A vertical setup for multiplying two binomials, (2x - 7) by (5x - 1), using a format similar to long multiplication, with the multiplication symbol and an underline indicating the operation.], alt: "A vertical setup for multiplying two binomials, (2x - 7) by (5x - 1), using a format similar to long multiplication, with the multiplication symbol and an underline indicating the operation.", caption: none)]),
  [Multiply #math.equation(block: false, alt: "2 x minus 7")[$2 x − 7$] by #math.equation(block: false, alt: "−1")[$−1$].], [#figure(figph[The image shows the expression '-2x + 7' in red, followed by the words 'partial product' in teal.], alt: "The image shows the expression '-2x + 7' in red, followed by the words 'partial product' in teal.", caption: none)],
  [Multiply #math.equation(block: false, alt: "2 x minus 7")[$2 x − 7$] by #math.equation(block: false, alt: "5 x")[$5 x$].], [#figure(figph[The expression 10x^2 - 35x is presented as a 'partial product,' representing an intermediate step in a larger polynomial multiplication or division calculation.], alt: "The expression 10x^2 - 35x is presented as a 'partial product,' representing an intermediate step in a larger polynomial multiplication or division calculation.", caption: none)],
  [Add like terms.], [#figure(figph[A quadratic expression 10x^2 - 37x + 7 is shown next to the word 'product'.], alt: "A quadratic expression 10x^2 - 37x + 7 is shown next to the word 'product'.", caption: none)],
))

Notice the partial products are the same as the terms in the FOIL method.

#figure(figph[On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.], alt: "On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the vertical method: #math.equation(block: false, alt: "open parenthesis 4 m minus 9 close parenthesis open parenthesis 3 m minus 7 close parenthesis .")[$( 4 m − 9 ) ( 3 m − 7 ) .$]

#solutionbox[
12#emph[m]#super[2] − 55#emph[m] + 63
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the vertical method: #math.equation(block: false, alt: "open parenthesis 6 n minus 5 close parenthesis open parenthesis 7 n minus 2 close parenthesis .")[$( 6 n − 5 ) ( 7 n − 2 ) .$]

#solutionbox[
42#emph[n]#super[2] − 47#emph[n] + 10
]
]

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplying Two Binomials]

- Distributive Property
- FOIL Method
- Vertical Method
]


=== Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the #strong[Distributive Property] or the #strong[Vertical Method]. We first look at an example using the Distributive Property.

#examplebox("Example 16")[][
Multiply using the Distributive Property: #math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis 2 x squared minus 5 x plus 8 close parenthesis .")[$( x + 3 ) ( 2 x^(2) − 5 x + 8 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Diagram depicting the initial steps of multiplying two polynomials: (x+3) and (2x^2-5x+8), highlighting the distribution of 'x' to the first two terms.], alt: "Diagram depicting the initial steps of multiplying two polynomials: (x+3) and (2x^2-5x+8), highlighting the distribution of 'x' to the first two terms.", caption: none)]),
  [Distribute.], [#figure(figph[An algebraic expression showing x(2x^2 - 5x + 8) + 3(2x^2 - 5x + 8), which represents a sum of terms suitable for factoring or expansion.], alt: "An algebraic expression showing x(2x^2 - 5x + 8) + 3(2x^2 - 5x + 8), which represents a sum of terms suitable for factoring or expansion.", caption: none)],
  [Multiply.], [#math.equation(block: false, alt: "2 x cubed minus 5 x squared plus 8 x plus 6 x squared minus 15 x plus 24")[$2 x^(3) − 5 x^(2) + 8 x + 6 x^(2) − 15 x + 24$]],
  [Combine like terms.], [#math.equation(block: false, alt: "2 x cubed plus x squared minus 7 x plus 24")[$2 x^(3) + x^(2) − 7 x + 24$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Distributive Property: #math.equation(block: false, alt: "open parenthesis y minus 1 close parenthesis open parenthesis y squared minus 7 y plus 2 close parenthesis .")[$( y − 1 ) ( y^(2) − 7 y + 2 ) .$]

#solutionbox[
#emph[y]#super[3] − 8#emph[y]#super[2] + 9#emph[y] − 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Distributive Property: #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis 3 x squared minus 4 x plus 5 close parenthesis .")[$( x + 2 ) ( 3 x^(2) − 4 x + 5 ) .$]

#solutionbox[
3#emph[x]#super[3] + 2#emph[x]#super[2] − 3#emph[x] + 10
]
]

Now let's do this same multiplication using the Vertical Method.

#examplebox("Example 17")[][
Multiply using the Vertical Method: #math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis 2 x squared minus 5 x plus 8 close parenthesis .")[$( x + 3 ) ( 2 x^(2) − 5 x + 8 ) .$]

#solutionbox[
It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A vertical long multiplication setup for polynomials. The top polynomial is 2x^2 - 5x + 8, and the bottom polynomial is x + 3, with a multiplication symbol on the left.], alt: "A vertical long multiplication setup for polynomials. The top polynomial is 2x^2 - 5x + 8, and the bottom polynomial is x + 3, with a multiplication symbol on the left.", caption: none)]),
  [Multiply #math.equation(block: false, alt: "open parenthesis 2 x squared minus 5 x plus 8 close parenthesis")[$( 2 x^(2) − 5 x + 8 )$] by 3.], [#figure(figph[The image shows the mathematical expression 6x^2 - 15x + 24.], alt: "The image shows the mathematical expression 6x^2 - 15x + 24.", caption: none)],
  [Multiply #math.equation(block: false, alt: "open parenthesis 2 x squared minus 5 x plus 8 close parenthesis")[$( 2 x^(2) − 5 x + 8 )$] by #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical expression, 2x^3 - 5x^2 + 8x, is displayed above a horizontal line, representing a fraction or a numerator.], alt: "A mathematical expression, 2x^3 - 5x^2 + 8x, is displayed above a horizontal line, representing a fraction or a numerator.", caption: none)],
  [Add like terms.], [#figure(figph[The mathematical expression 2x^3 + x^2 - 7x + 24 is displayed in black font against a white background.], alt: "The mathematical expression 2x^3 + x^2 - 7x + 24 is displayed in black font against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Vertical Method: #math.equation(block: false, alt: "open parenthesis y minus 1 close parenthesis open parenthesis y squared minus 7 y plus 2 close parenthesis .")[$( y − 1 ) ( y^(2) − 7 y + 2 ) .$]

#solutionbox[
#emph[y]#super[3] − 8#emph[y]#super[2] + 9#emph[y] − 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply using the Vertical Method: #math.equation(block: false, alt: "open parenthesis x plus 2 close parenthesis open parenthesis 3 x squared minus 4 x plus 5 close parenthesis .")[$( x + 2 ) ( 3 x^(2) − 4 x + 5 ) .$]

#solutionbox[
3#emph[x]#super[3] + 2#emph[x]#super[2] − 3#emph[x] + 10
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24multmonomials")[Multiply Monomials]
- #link("https://www.openstax.org/l/24multpolyns")[Multiply Polynomials]
- #link("https://www.openstax.org/l/24multpolyns2")[Multiply Polynomials #math.equation(block: false, alt: "2")[$2$]]
- #link("https://www.openstax.org/l/24multpolynsrev")[Multiply Polynomials Review]
- #link("https://www.openstax.org/l/24multpolynsdis")[Multiply Polynomials Using the Distributive Property]
- #link("https://www.openstax.org/l/24multbinomials")[Multiply Binomials]
]


=== Key Concepts

- #emph[Use the FOIL method for multiplying two binomials.]
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. Multiply the #emph[First] terms.], [#figure(figph[A diagram illustrating the 'first', 'outer', 'inner', and 'last' (FOIL) terms when multiplying two binomials, (a+b)(c+d).], alt: "A diagram illustrating the 'first', 'outer', 'inner', and 'last' (FOIL) terms when multiplying two binomials, (a+b)(c+d).", caption: none)]),
  [Step 2. Multiply the #emph[Outer] terms.], [],
  [Step 3. Multiply the #emph[Inner] terms.], [],
  [Step 4. Multiply the #emph[Last] terms.], [],
  [Step 5. Combine like terms, when possible.], [],
))
- #emph[Multiplying Two Binomials:] To multiply binomials, use the:
- Distributive Property
- FOIL Method
- Vertical Method
- #emph[Multiplying a Trinomial by a Binomial:] To multiply a trinomial by a binomial, use the:
- Distributive Property
- Vertical Method


==== Practice Makes Perfect

#strong[Multiply a Polynomial by a Monomial]

In the following exercises, multiply.

#math.equation(block: true, alt: "4 open parenthesis x plus 10 close parenthesis")[$4 ( x + 10 )$]

#solutionbox[
4#emph[x] + 40
]

#math.equation(block: true, alt: "6 open parenthesis y plus 8 close parenthesis")[$6 ( y + 8 )$]

#math.equation(block: true, alt: "15 open parenthesis r minus 24 close parenthesis")[$15 ( r − 24 )$]

#solutionbox[
15#emph[r] − 360
]

#math.equation(block: true, alt: "12 open parenthesis v minus 30 close parenthesis")[$12 ( v − 30 )$]

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

#solutionbox[
−3#emph[m] − 33
]

#math.equation(block: true, alt: "−4 open parenthesis p plus 15 close parenthesis")[$−4 ( p + 15 )$]

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

#solutionbox[
−8#emph[z] + 40
]

#math.equation(block: true, alt: "−3 open parenthesis x minus 9 close parenthesis")[$−3 ( x − 9 )$]

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

#solutionbox[
#emph[u]#super[2] + 5#emph[u]
]

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

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

#solutionbox[
#emph[n]#super[3] − 3#emph[n]#super[2]
]

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

#math.equation(block: true, alt: "12 x open parenthesis x minus 10 close parenthesis")[$12 x ( x − 10 )$]

#solutionbox[
12#emph[x]#super[2] − 120#emph[x]
]

#math.equation(block: true, alt: "9 m open parenthesis m minus 11 close parenthesis")[$9 m ( m − 11 )$]

#math.equation(block: true, alt: "−9 a open parenthesis 3 a plus 5 close parenthesis")[$−9 a ( 3 a + 5 )$]

#solutionbox[
−27#emph[a]#super[2] − 45#emph[a]
]

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

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

#solutionbox[
24#emph[x]#super[2] + 6#emph[xy]
]

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

#math.equation(block: true, alt: "5 p open parenthesis 11 p minus 5 q close parenthesis")[$5 p ( 11 p − 5 q )$]

#solutionbox[
55#emph[p]#super[2] − 25#emph[pq]
]

#math.equation(block: true, alt: "12 u open parenthesis 3 u minus 4 v close parenthesis")[$12 u ( 3 u − 4 v )$]

#math.equation(block: true, alt: "3 open parenthesis v squared plus 10 v plus 25 close parenthesis")[$3 ( v^(2) + 10 v + 25 )$]

#solutionbox[
3#emph[v]#super[2] + 30#emph[v] + 75
]

#math.equation(block: true, alt: "6 open parenthesis x squared plus 8 x plus 16 close parenthesis")[$6 ( x^(2) + 8 x + 16 )$]

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

#solutionbox[
8#emph[n]#super[3] − 8#emph[n]#super[2] + 2#emph[n]
]

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

#math.equation(block: true, alt: "−8 y open parenthesis y squared plus 2 y minus 15 close parenthesis")[$−8 y ( y^(2) + 2 y − 15 )$]

#solutionbox[
−8#emph[y]#super[3] − 16#emph[y]#super[2] + 120#emph[y]
]

#math.equation(block: true, alt: "−5 m open parenthesis m squared plus 3 m minus 18 close parenthesis")[$−5 m ( m^(2) + 3 m − 18 )$]

#math.equation(block: true, alt: "5 q cubed open parenthesis q squared minus 2 q plus 6 close parenthesis")[$5 q^(3) ( q^(2) − 2 q + 6 )$]

#solutionbox[
5#emph[q]#super[5] − 10#emph[q]#super[4] + 30#emph[q]#super[3]
]

#math.equation(block: true, alt: "9 r cubed open parenthesis r squared minus 3 r plus 5 close parenthesis")[$9 r^(3) ( r^(2) − 3 r + 5 )$]

#math.equation(block: true, alt: "−4 z squared open parenthesis 3 z squared plus 12 z minus 1 close parenthesis")[$−4 z^(2) ( 3 z^(2) + 12 z − 1 )$]

#solutionbox[
−12#emph[z]#super[4] − 48#emph[z]#super[3] + 4#emph[z]#super[2]
]

#math.equation(block: true, alt: "−3 x squared open parenthesis 7 x squared plus 10 x minus 1 close parenthesis")[$−3 x^(2) ( 7 x^(2) + 10 x − 1 )$]

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

#solutionbox[
2#emph[y]#super[2] − 9#emph[y]
]

#math.equation(block: true, alt: "open parenthesis 8 b minus 1 close parenthesis b")[$( 8 b − 1 ) b$]

#math.equation(block: true, alt: "open parenthesis w minus 6 close parenthesis times 8")[$( w − 6 ) #h(0.2em) · #h(0.2em) 8$]

#solutionbox[
8#emph[w] − 48
]

#math.equation(block: true, alt: "open parenthesis k minus 4 close parenthesis times 5")[$( k − 4 ) #h(0.2em) · #h(0.2em) 5$]

#strong[Multiply a Binomial by a Binomial]

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method

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

#solutionbox[
#emph[x]#super[2] + 10#emph[x] + 24
]

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

#math.equation(block: true, alt: "open parenthesis n plus 12 close parenthesis open parenthesis n minus 3 close parenthesis")[$( n + 12 ) ( n − 3 )$]

#solutionbox[
#emph[n]#super[2] + 9#emph[n] − 36
]

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

In the following exercises, multiply the following binomials. Use any method.

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

#solutionbox[
#emph[y]#super[2] + 11#emph[y] + 24
]

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

#math.equation(block: true, alt: "open parenthesis a plus 6 close parenthesis open parenthesis a plus 16 close parenthesis")[$( a + 6 ) ( a + 16 )$]

#solutionbox[
#emph[a]#super[2] + 22#emph[a] + 96
]

#math.equation(block: true, alt: "open parenthesis q plus 8 close parenthesis open parenthesis q plus 12 close parenthesis")[$( q + 8 ) ( q + 12 )$]

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

#solutionbox[
#emph[u]#super[2] − 14#emph[u] + 45
]

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

#math.equation(block: true, alt: "open parenthesis z minus 10 close parenthesis open parenthesis z minus 22 close parenthesis")[$( z − 10 ) ( z − 22 )$]

#solutionbox[
#emph[z]#super[2] − 32#emph[z] + 220
]

#math.equation(block: true, alt: "open parenthesis b minus 5 close parenthesis open parenthesis b minus 24 close parenthesis")[$( b − 5 ) ( b − 24 )$]

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

#solutionbox[
#emph[x]#super[2] + 3#emph[x] − 28
]

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

#math.equation(block: true, alt: "open parenthesis v plus 12 close parenthesis open parenthesis v minus 5 close parenthesis")[$( v + 12 ) ( v − 5 )$]

#solutionbox[
#emph[v]#super[2] + 7#emph[v] − 60
]

#math.equation(block: true, alt: "open parenthesis d plus 15 close parenthesis open parenthesis d minus 4 close parenthesis")[$( d + 15 ) ( d − 4 )$]

#math.equation(block: true, alt: "open parenthesis 6 n plus 5 close parenthesis open parenthesis n plus 1 close parenthesis")[$( 6 n + 5 ) ( n + 1 )$]

#solutionbox[
6#emph[n]#super[2] + 11#emph[n] + 5
]

#math.equation(block: true, alt: "open parenthesis 7 y plus 1 close parenthesis open parenthesis y plus 3 close parenthesis")[$( 7 y + 1 ) ( y + 3 )$]

#math.equation(block: true, alt: "open parenthesis 2 m minus 9 close parenthesis open parenthesis 10 m plus 1 close parenthesis")[$( 2 m − 9 ) ( 10 m + 1 )$]

#solutionbox[
20#emph[m]#super[2] − 88#emph[m] − 9
]

#math.equation(block: true, alt: "open parenthesis 5 r minus 4 close parenthesis open parenthesis 12 r plus 1 close parenthesis")[$( 5 r − 4 ) ( 12 r + 1 )$]

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

#solutionbox[
16#emph[c]#super[2] − 1
]

#math.equation(block: true, alt: "open parenthesis 8 n minus 1 close parenthesis open parenthesis 8 n plus 1 close parenthesis")[$( 8 n − 1 ) ( 8 n + 1 )$]

#math.equation(block: true, alt: "open parenthesis 3 u minus 8 close parenthesis open parenthesis 5 u minus 14 close parenthesis")[$( 3 u − 8 ) ( 5 u − 14 )$]

#solutionbox[
15#emph[u]#super[2] − 82#emph[u] + 112
]

#math.equation(block: true, alt: "open parenthesis 2 q minus 5 close parenthesis open parenthesis 7 q minus 11 close parenthesis")[$( 2 q − 5 ) ( 7 q − 11 )$]

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis open parenthesis 2 a plus 3 b close parenthesis")[$( a + b ) ( 2 a + 3 b )$]

#solutionbox[
2#emph[a]#super[2] + 5#emph[ab] + 3#emph[b]#super[2]
]

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

#math.equation(block: true, alt: "open parenthesis 5 x minus y close parenthesis open parenthesis x minus 4 close parenthesis")[$( 5 x − y ) ( x − 4 )$]

#solutionbox[
5#emph[x]#super[2] − 20#emph[x] − #emph[xy] + 4#emph[y]
]

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

#strong[Multiply a Trinomial by a Binomial]

In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method.

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

#solutionbox[
#emph[u]#super[3] + 7#emph[u]#super[2] + 14#emph[u] + 8
]

#math.equation(block: true, alt: "open parenthesis x plus 5 close parenthesis open parenthesis x squared plus 8 x plus 3 close parenthesis")[$( x + 5 ) ( x^(2) + 8 x + 3 )$]

#math.equation(block: true, alt: "open parenthesis a plus 10 close parenthesis open parenthesis 3 a squared plus a minus 5 close parenthesis")[$( a + 10 ) ( 3 a^(2) + a − 5 )$]

#solutionbox[
3#emph[a]#super[3] + 31#emph[a]#super[2] + 5#emph[a] − 50
]

#math.equation(block: true, alt: "open parenthesis n plus 8 close parenthesis open parenthesis 4 n squared plus n minus 7 close parenthesis")[$( n + 8 ) ( 4 n^(2) + n − 7 )$]

In the following exercises, multiply. Use either method.

#math.equation(block: true, alt: "open parenthesis y minus 6 close parenthesis open parenthesis y squared minus 10 y plus 9 close parenthesis")[$( y − 6 ) ( y^(2) − 10 y + 9 )$]

#solutionbox[
#emph[y]#super[3] − 16#emph[y]#super[2] + 69#emph[y] − 54
]

#math.equation(block: true, alt: "open parenthesis k minus 3 close parenthesis open parenthesis k squared minus 8 k plus 7 close parenthesis")[$( k − 3 ) ( k^(2) − 8 k + 7 )$]

#math.equation(block: true, alt: "open parenthesis 2 x plus 1 close parenthesis open parenthesis x squared minus 5 x minus 6 close parenthesis")[$( 2 x + 1 ) ( x^(2) − 5 x − 6 )$]

#solutionbox[
2#emph[x]#super[3] − 9#emph[x]#super[2] − 17#emph[x] − 6
]

#math.equation(block: true, alt: "open parenthesis 5 v plus 1 close parenthesis open parenthesis v squared minus 6 v minus 10 close parenthesis")[$( 5 v + 1 ) ( v^(2) − 6 v − 10 )$]


==== Everyday Math

#strong[Mental math] You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply #math.equation(block: false, alt: "13")[$13$] times #math.equation(block: false, alt: "15 .")[$15 .$] Think of #math.equation(block: false, alt: "13")[$13$] as #math.equation(block: false, alt: "10 plus 3")[$10 + 3$] and #math.equation(block: false, alt: "15")[$15$] as #math.equation(block: false, alt: "10 plus 5 .")[$10 + 5 .$]

+ ⓐ Multiply #math.equation(block: false, alt: "open parenthesis 10 plus 3 close parenthesis open parenthesis 10 plus 5 close parenthesis")[$( 10 + 3 ) ( 10 + 5 )$] by the FOIL method.
+ ⓑ Multiply #math.equation(block: false, alt: "13 times 15")[$13 #h(0.2em) · #h(0.2em) 15$] without using a calculator.
+ ⓒ Which way is easier for you? Why?

#solutionbox[
+ ⓐ 195
+ ⓑ 195
+ ⓒ Answers will vary.
]

#strong[Mental math] You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply #math.equation(block: false, alt: "18")[$18$] times #math.equation(block: false, alt: "17 .")[$17 .$] Think of #math.equation(block: false, alt: "18")[$18$] as #math.equation(block: false, alt: "20 minus 2")[$20 − 2$] and #math.equation(block: false, alt: "17")[$17$] as #math.equation(block: false, alt: "20 minus 3 .")[$20 − 3 .$]

+ ⓐ Multiply #math.equation(block: false, alt: "open parenthesis 20 minus 2 close parenthesis open parenthesis 20 minus 3 close parenthesis")[$( 20 − 2 ) ( 20 − 3 )$] by the FOIL method.
+ ⓑ Multiply #math.equation(block: false, alt: "18 times 17")[$18 #h(0.2em) · #h(0.2em) 17$] without using a calculator.
+ ⓒ Which way is easier for you? Why?


==== Writing Exercises

Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why?

#solutionbox[
Answers will vary.
]

Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?


==== Self Check

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

#figure(figph[A self-assessment table for students to evaluate their understanding of multiplying polynomials, binomials, and trinomials, with options to rate their skill level as 'Confidently,' 'With some help,' or 'No-I don't get it!'], alt: "A self-assessment table for students to evaluate their understanding of multiplying polynomials, binomials, and trinomials, with options to rate their skill level as 'Confidently,' 'With some help,' or 'No-I don't get it!'", caption: none)

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