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Polynomials are useful for modeling different real-life situations. Sometimes more than one polynomial could be involved and the polynomials may need to be multiplied. For such a reason, this lesson aims to teach different methods for multiplying polynomials.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

When two polynomials are added or subtracted, the result is also a polynomial. What about the product of two polynomials? Consider, for example, the following pair of polynomials.

$P(x)Q(x) =2x_{3}−x+5=4(x+2)(x−1) $

Is it possible to calculate the product of $P(x)$ and $Q(x)?$ Is $P(x)⋅Q(x)$ also a polynomial? In the affirmative case, what are the degree, the leading coefficient, and the constant term of the resulting polynomial? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Degree <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["5"]}}

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Constant Term <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-40"]}}

Discussion

Consider a polynomial $P(x)$ written in standard form.
Note that the right-hand side is a polynomial of the same degree as $P(x)$ — the degree of both is $n.$ On the other hand, if $C =1,$ the leading coefficient of $C⋅P(x)$ is different from the leading coefficient of $P(x).$

Consequently, when a polynomial $P(x)$ is multiplied by a constant $C,$ the degree of the polynomial does not change and the leading coefficient equals $C$ times the leading coefficient of $P(x).$

$P(x)=a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{1}x+a_{0} $

Using the Distributive Property, multiply $P(x)$ by a non-zero constant $C.$
$P(x)=a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{1}x+a_{0}$

MultEqn

$LHS⋅C=RHS⋅C$

$C⋅P(x)=C(a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{1}x+a_{0})$

Distr

Distribute $C$

$C⋅P(x)=C⋅a_{n}x_{n}+C⋅a_{n−1}x_{n−1}+⋯+C⋅a_{1}x+C⋅a_{0}$

Polynomial | Degree | Leading Coefficient |
---|---|---|

$P(x)=a_{n}x_{n}+a_{n−1}x_{n−1}+⋯+a_{1}x+a_{0}$ | $n$ | $a_{n}$ |

$C⋅P(x)=C⋅a_{n}x_{n}+C⋅a_{n−1}x_{n−1}+⋯+C⋅a_{1}x+C⋅a_{0}$ | $n$ | $C⋅a_{n}$ |

Example

Ali is on his school's student council. He is in charge of decorations for an upcoming school dance. He had the following chat with Kevin.
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### Hint

### Solution

The area of the poster will be $112$ square feet.

After the talk, Kevin made the following diagram.

a Write a polynomial $A(x),$ in standard form, that models the area of the poster.

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b What are the degree and leading coefficient of $A(x)?$

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c The next day, Ali called Kevin and told him that the wall is $16$ feet wide. What area will the poster have?

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a The poster is $2$ feet shorter and narrower than the wall. To find its area, multiply the length by the width.

b The degree of a polynomial is the greatest exponent in the polynomial. The leading coefficient is the coefficient in front of the term with the greatest exponent.

c Evaluate $A(x)$ at $x=16.$

a The area of the poster equals its length multiplied by its width.

$A=ℓ⋅w $

From the chat and the diagram made by Kevin, the poster has to be $2$ feet shorter and narrower than the wall. Since the wall is $10$ feet tall, the poster will be $8$ feet tall. The wall is $x$ feet wide, so the width of the poster is $x−2$ feet.
A polynomial modeling the area of the poster is obtained by substituting the poster's dimensions into the formula for the area.
$A(x)=(x−2)⋅8⇓A(x)=8x−16 $

Note that this polynomial gives the area of the poster depending on the width of the wall.
b Start by writing the polynomial found in Part A. Then, highlight the term with the greatest exponent and its coefficient.

$A(x)=8x−16⇕A(x)=8x_{1}−16 $

The greatest exponent is $1,$ which means that the degree of the polynomial is $1.$ The coefficient in front of the term with the greatest exponent is $8.$ Thus, the leading coefficient is $8.$
c Since $x$ represents the width of the wall, to find the area of the poster, substitute $x=16$ into $A(x).$

$A(x)=8x−16$

Substitute

$x=16$

$A(16)=8(16)−16$

Multiply

Multiply

$A(16)=128−16$

SubTerms

Subtract terms

$A(16)=112$

Discussion

Different methods can be used to multiply two polynomials. The following three methods are based on the Distributive Property.

Method

The product of two polynomials can be calculated by using the Distributive Property. Consider, for example, the following pair of polynomials.
*expand_more*
*expand_more*
*expand_more*
*expand_more*
Note that multiplying a polynomial with $n$ terms by a polynomial with $m$ terms produces $n⋅m$ products. When two polynomials are multiplied, the product is a new polynomial whose degree equals the sum of the degrees of the polynomial factors.

$P(x)Q(x) =x_{3}+2x_{2}−3=x_{2}+4 $

To multiply these two polynomials, the following four steps can be followed.
1

Distribute One Polynomial to All the Terms of the Other

Start by writing the product $P(x)⋅Q(x).$

$(x_{3}+2x_{2}−3)(x_{2}+4) $

Next, distribute $P(x)$ to each term of $Q(x).$
$(x_{3}+2x_{2}−3)(x_{2}+4)$

Distr

Distribute $x_{3}+2x_{2}−3$

$(x_{3}+2x_{2}−3)x_{2}+(x_{3}+2x_{2}−3)4$

2

Clear Parentheses by Applying the Distributive Property

3

Apply the Product of Powers Property

Use the Product of Powers Property to rewrite the product of two or more exponents as a single power.

4

Combine Like Terms and Simplify

Finally, combine like terms and perform all the required operations to simplify the result.

$(x_{5}+2x_{4}−3⋅x_{2})+(x_{3}⋅4+2x_{2}⋅4−3⋅4)$

Multiply

Multiply

$(x_{5}+2x_{4}−3x_{2})+(4x_{3}+8x_{2}−12)$

CommutativePropAdd

Commutative Property of Addition

$x_{5}+2x_{4}+4x_{3}−3x_{2}+8x_{2}−12$

AssociativePropAdd

Associative Property of Addition

$x_{5}+2x_{4}+4x_{3}+(-3x_{2}+8x_{2})−12$

AddTerms

Add terms

$x_{5}+2x_{4}+4x_{3}+5x_{2}−12$

Method

Given two polynomials, their product can be calculated by using a box or table. Consider, for example, the following pair of polynomials.
*expand_more*

*expand_more*
*expand_more*
*expand_more*

$P(x)Q(x) =x_{3}+2x_{2}−3=x_{2}+4 $

To multiply these two polynomials, the following four steps can be followed.
1

Determine the Dimensions of the Table

Start by drawing a table that has as many rows as there are terms in the first polynomial and that has as many columns as there are terms in the second polynomial.

Polynomial | Number of Terms |
---|---|

$P(x)=x_{3}+2x_{2}−3$ | $3$ |

$Q(x)=x_{2}+4$ | $2$ |

For example, a table with $3$ rows and $2$ columns is needed to multiply $P(x)$ by $Q(x).$

2

Write the Row and Column Labels of the Table

Now, write each term of the first polynomial at the left of each cell of the first column. Similarly, write each term of the second polynomial above each cell of the first row.

3

Multiply the Terms to Fill the Table

Next, fill in the table's cells by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of $x_{3}$ and $x_{2}.$ The remaining cells can be filled by following the same procedure.

4

Add Terms and Simplify

Finally, add all the expressions inside the table and combine like terms, if any.

The product of these polynomials has been found to be $x_{5}+2x_{4}+4x_{3}+5x_{2}−12.$

Method

The FOIL method is a mnemonic for remembering how to multiply two binomials. The word **FOIL** is an acronym for the words $First,$ $Outer,$ $Inner,$ and $Last.$ Consider, for example, the following product.
*expand_more*
*expand_more*
*expand_more*
*expand_more*
*expand_more*
The following applet illustrates the FOIL method using two arbitrary binomials.

$(x+6)(3x−2) $

These two binomials can be multiplied by following five steps.
1

Multiply the First Terms

Start by multiplying the first terms of each binomial. In this case, multiply $x$ by $3x.$

$(x+6)(3x−2)=x(3x)−2x+18x−12 $

The empty box is there as a reminder that there are still missing terms. 2

Multiply the Outer Terms

Next, multiply the outer terms — that is, multiply the first term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply $x$ by $-2.$

$(x+6)(3x−2)=x(3x)+x(-2)18x−12 $

3

Multiply the Inner Terms

Now multiply the inner terms — that is, multiply the second term of the left-hand side binomial by the first term of the right-hand side binomial. In this case, multiply $6$ by $3x.$

$(x+6)(3x−2)=x(3x)+x(-2)+6(3x)−12 $

4

Multiply the Last Terms

Next, multiply the last terms of each binomial — that is, multiply the second term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply $6$ by $-2.$

$(x+6)(3x−2)=x(3x)+x(-2)+6(3x)+6(-2) $

5

Simplify

Finally, find each product and combine like terms, if any, to simplify the resulting expression.

$(x+6)(3x−2)=x(3x)+x(-2)+6(3x)+6(-2)$

CommutativePropMult

Commutative Property of Multiplication

$(x+6)(3x−2)=3x⋅x+x(-2)+6(3x)+6(-2)$

ProdToPowTwoFac

$a⋅a=a_{2}$

$(x+6)(3x−2)=3x_{2}+x(-2)+6(3x)+6(-2)$

MultPosNeg

$a(-b)=-a⋅b$

$(x+6)(3x−2)=3x_{2}−x(2)+6(3x)−6(2)$

Multiply

Multiply

$(x+6)(3x−2)=3x_{2}−2x+18x−12$

AddTerms

Add terms

$(x+6)(3x−2)=3x_{2}+16x−12$

Like any other polynomial multiplication, the FOIL method is based on the Distributive Property.

Example

Diego's parents recently bought a piece of land where they plan to raise pigs. They need to fence off a rectangular pigpen before buying the pigs. A farmer friend told Diego that the dimensions of the pigpen, in yards, vary according to the number of pigs $x$ that are being raised in it.

a Write a polynomial $A(x),$ in standard form, that models the pigpen area. Then, state the degree and the leading coefficient of $A(x).$

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b If Diego plans to raise $15$ pigs, what should be the area of the pigpen?

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a The area of a rectangle equals the length times the width.

b Evaluate $A(x)$ at $x=15.$

a The area of a rectangle is obtained by multiplying its length by its width. Start by writing the dimensions suggested by Diego's friend.

$Length:Width: 5x+23x+1 $

By multiplying the two binomials, a polynomial modeling the pigpen area will be obtained.
$A(x)=(5x+2)(3x+1) $

To write $A(x)$ in standard form, the product on the right-hand side can be performed by using the FOIL method.
Consequently, the degree of $A(x)$ is $2$ and its leading coefficient is $15.$

b To determine the area of a pigpen large enough to raise $15$ pigs, substitute $15$ for $x$ into $A(x)=15x_{2}+11x+2$ and simplify.

$A(x)=15x_{2}+11x+2$

Substitute

$x=15$

$A(15)=15(15)_{2}+11(15)+2$

CalcPowProd

Calculate power and product

$A(15)=15(225)+165+2$

Multiply

Multiply

$A(15)=3375+165+2$

AddTerms

Add terms

$A(15)=3542$

Example

Consider a right triangle on the coordinate plane with vertices $J(-10,0),$ $K(x,0),$ and $L(x,P(x)),$ where $P(x)=x_{3}+3x_{2}−18x+54.$
### Hint

### Solution

a Write a polynomial $A(x),$ in standard form, that represents the area of $△JKL.$ Then, state the degree and leading coefficient of $A(x).$

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b If $K$ has coordinates $(3,0),$ what is the area of $△JKL?$

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Area <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["351"]}}

c Which one of the following triangles has the larger area?

{"type":"choice","form":{"alts":["The triangle generated by <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">4<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>","The triangle generated by <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">K<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

a The area of a triangle equals half the product of its base and its height. Note that the height of $△JKL$ is $P(x)$ and the base is $x+10.$

b Evaluate $A(x)$ at $x=3.$

c Find and compare $A(-4)$ and $A(-1).$

a The area of a triangle is half the product of its base and its height. Since the given triangle is a right triangle, its area equals half the product of its legs.

$A=21 ⋅JK⋅KL $

The length of the legs can be written using the given expressions.
$JKKL =x+10=x_{3}+3x_{2}−18x+54 $

Next, substitute the previous expressions into the formula for the area.
$A(x)=21 (x+10)(x_{3}+3x_{2}−18x+54) $

For simplicity, the product of the two polynomials will be computed first. Then, the resulting polynomial will be multiplied by $21 .$ To perform the polynomial multiplication, the Box Method can be used. To do so, start by drawing a $2×4$ table and writing the row and column labels.
The next step is to fill the table by multiplying the terms written on the corresponding borders.
Now, add all the terms inside the table and simplify by combining like terms.

Having computed $JK⋅KL,$ a polynomial modeling the area of $△JKL$ can be found.

$A(x)=21 JK⋅KL$

Substitute

$JK⋅KL=x_{4}+13x_{3}+12x_{2}−126x+540$

$A(x)=21 (x_{4}+13x_{3}+12x_{2}−126x+540)$

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