3. Multiplying Polynomials
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Chapter 7
3.
Multiplying Polynomials
This lesson offers a comprehensive guide on how to multiply polynomials using various techniques, including the distributive property and the FOIL method. It goes beyond mere definitions to explore real-world applications. For example, it discusses how these methods can be used in calculating areas and volumes in geometry, or in solving complex equations in algebra. The FOIL method, which stands for First, Outer, Inner, Last, is explained as a mnemonic to help remember the steps involved in multiplying binomials. The distributive property is also elaborated upon, showing how it can simplify complex polynomial equations. Overall, this material serves as an invaluable lesson for anyone looking to master polynomial multiplication and its applications in different fields of mathematics.
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Lesson Settings & Tools
| | 11 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Multiplying Polynomials
Slide of 11
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
Investigating the Product of Polynomials
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)=2x3−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":[]},"rendered":false},"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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Discussion
Multiplying a Polynomial by a Constant
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)=anxn+an−1xn−1+⋯+a1x+a0
Using the Distributive Property, multiply P(x) by a non-zero constant C.
P(x)=anxn+an−1xn−1+⋯+a1x+a0
MultEqn
LHS⋅C=RHS⋅C
C⋅P(x)=C(anxn+an−1xn−1+⋯+a1x+a0)
Distr
Distribute C
C⋅P(x)=C⋅anxn+C⋅an−1xn−1+⋯+C⋅a1x+C⋅a0
| Polynomial | Degree | Leading Coefficient |
|---|---|---|
| P(x)=anxn+an−1xn−1+⋯+a1x+a0 | n | an |
| C⋅P(x)=C⋅anxn+C⋅an−1xn−1+⋯+C⋅a1x+C⋅a0 | n | C⋅an |
Example
Finding the Area of a Poster
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.
After the talk, Kevin made the following diagram. 
A polynomial modeling the area of the poster is obtained by substituting the poster's dimensions into the formula for the area.
The area of the poster will be 112 square feet.
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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Hint
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.
Solution
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(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)=8x1−16
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
Methods for Multiplying Polynomials
Different methods can be used to multiply two polynomials. The following three methods are based on the Distributive Property.
Method
Multiplying Polynomials Using the Distributive Property
The product of two polynomials can be calculated by using the Distributive Property. Consider, for example, the following pair of polynomials.
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)=x3+2x2−3=x2+4
To multiply these two polynomials, the following four steps can be followed.
1
Distribute One Polynomial to All the Terms of the Other
expand_more Start by writing the product P(x)⋅Q(x).
(x3+2x2−3)(x2+4)
Next, distribute P(x) to each term of Q(x).
2
Clear Parentheses by Applying the Distributive Property
expand_more 3
Apply the Product of Powers Property
expand_more 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
expand_more Finally, combine like terms and perform all the required operations to simplify the result.
(x5+2x4−3⋅x2)+(x3⋅4+2x2⋅4−3⋅4)
Multiply
Multiply
(x5+2x4−3x2)+(4x3+8x2−12)
CommutativePropAdd
Commutative Property of Addition
x5+2x4+4x3−3x2+8x2−12
AssociativePropAdd
Associative Property of Addition
x5+2x4+4x3+(-3x2+8x2)−12
AddTerms
Add terms
x5+2x4+4x3+5x2−12
Method
Multiplying Polynomials Using the Box Method
Given two polynomials, their product can be calculated by using a box or table. Consider, for example, the following pair of polynomials.
P(x)Q(x)=x3+2x2−3=x2+4
To multiply these two polynomials, the following four steps can be followed.
1
Determine the Dimensions of the Table
expand_more 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)=x3+2x2−3 | 3 |
| Q(x)=x2+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
expand_more 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
expand_more 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 x3 and x2. The remaining cells can be filled by following the same procedure. 
4
Add Terms and Simplify
expand_more Finally, add all the expressions inside the table and combine like terms, if any.
The product of these polynomials has been found to be x5+2x4+4x3+5x2−12.
Method
The FOIL 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.
The following applet illustrates the FOIL method using two arbitrary binomials.
Like any other polynomial multiplication, the FOIL method is based on the Distributive Property.
(x+6)(3x−2)
These two binomials can be multiplied by following five steps.
1
Multiply the First Terms
expand_more 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
expand_more 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
expand_more 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
expand_more 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
expand_more 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=a2
(x+6)(3x−2)=3x2+x(-2)+6(3x)+6(-2)
MultPosNeg
a(-b)=-a⋅b
(x+6)(3x−2)=3x2−x(2)+6(3x)−6(2)
Multiply
Multiply
(x+6)(3x−2)=3x2−2x+18x−12
AddTerms
Add terms
(x+6)(3x−2)=3x2+16x−12
Example
Modeling the Pigpen Area
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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Hint
a The area of a rectangle equals the length times the width.
b Evaluate A(x) at x=15.
Solution
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.
b To determine the area of a pigpen large enough to raise 15 pigs, substitute 15 for x into A(x)=15x2+11x+2 and simplify.
A(x)=15x2+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
Determining the Area of a Triangle
Consider a right triangle on the coordinate plane with vertices J(-10,0), K(x,0), and L(x,P(x)), where P(x)=x3+3x2−18x+54.
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.
Consequently, the degree of A(x) is 4 and its leading coefficient is 21.
To find the area of the above triangle, substitute 3 for x into the polynomial A(x) found in Part A.
In consequence, when K has coordinates (3,0), the triangle JKL has an area of 351.
To determine which triangle has the larger area, find A(-4) and A(-1) and compare them. Start by finding A(-4).
Next, find the value of A(-1).
As can be seen, A(-1) is greater than A(-4). Therefore, the triangle generated by K2(-1,0) has the larger area.
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?
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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}
Hint
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).
Solution
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=x3+3x2−18x+54
Next, substitute the previous expressions into the formula for the area.
A(x)=21(x+10)(x3+3x2−18x+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. A(x)=21JK⋅KL
Substitute
JK⋅KL=x4+13x3+12x2−126x+540
A(x)=21(x4+13x3+12x2−126x+540)
Distr
Distribute 21
A(x)=21x4+213x3+6x2−63x+270
b If the coordinates of K are (3,0), it implies that the coordinates of L are (3,P(3)).
A(x)=21x4+213x3+6x2−63x+270
▼
Substitute 3 for x and simplify
Substitute
x=3
A(3)=21(3)4+213(3)3+6(3)2−63(3)+270
CalcPowProd
Calculate power and product
A(3)=281+2351+54−189+270
AddFrac
Add fractions
A(3)=281+351+54−189+270
AddSubTerms
Add and subtract terms
A(3)=2432+135
CalcQuot
Calculate quotient
A(3)=216+135
AddTerms
Add terms
A(3)=351
c The triangles generated by K1 and K2 look as follows.
A(x)=21x4+213x3+6x2−63x+270
▼
Substitute -4 for x and simplify
Substitute
x=-4
A(-4)=21(-4)4+213(-4)3+6(-4)2−63(-4)+270
CalcPowProd
Calculate power and product
A(-4)=2256−2832+96+252+270
CalcQuot
Calculate quotient
A(-4)=128−416+96+252+270
AddSubTerms
Add and subtract terms
A(-4)=330
A(x)=21x4+213x3+6x2−63x+270
▼
Substitute -1 for x and simplify
Substitute
x=-1
A(-1)=21(-1)4+213(-1)3+6(-1)2−63(-1)+270
CalcPowProd
Calculate power and product
A(-1)=21−213+6+63+270
AddFrac
Add fractions
A(-1)=21−13+6+63+270
SubTerms
Subtract terms
A(-1)=-212+6+63+270
CalcQuot
Calculate quotient
A(-1)=-6+6+63+270
AddSubTerms
Add and subtract terms
A(-1)=333
Example
Determining a Toy's Volume
Izabella bought her nephew a bag of magic grow toys for his birthday. Among the toys, there was a trailer and its rectangular container. Initially, the container was 5 centimeters long, 2 centimeters wide, and 3 centimeters high.
ℓ(t)w(t)h(t)=41t2+5=-91t2+2t+2=t+3
Here, t represents the number of hours the toy has been underwater. a Write a polynomial V(t), in standard form, that represents the volume of the container after being underwater for t hours.
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b Izabella's nephew placed the toy underwater at 10:00 AM and took it out at 4:00 PM. What is the volume of the container now?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Volume <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":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8238736249999999em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord Roboto-Regular\">cm<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["1260"]}}
Hint
a The volume of a rectangular container is obtained by multiplying its three dimensions.
b How many hours was the container underwater? Evaluate V(t) at this value.
Solution
a The volume of a rectangular container is found by multiplying its three dimensions.
V=w⋅ℓ⋅h
Therefore, to find a polynomial representing the container's volume after passing t hours underwater, multiply the expressions that model the dimensions of the container.
w(t)⋅ℓ(t)⋅h(t)⇓(41t2+5)(-91t2+2t+2)(t+3)
By applying the Commutative Property of Multiplication, the previous product can be rewritten to have the two binomials next to each other.
(-91t2+2t+2)(41t2+5)(t+3)
Then, these two binomials can be multiplied using the FOIL method. For simplicity, the trinomial will not be written during this multiplication. - Start by multiplying the first terms.(41t2+5)(t+3)=41t2⋅t+⋯
- Continue by multiplying the outer terms.(41t2+5)(t+3)=41t2⋅t+41t2⋅3+⋯
- Next, multiply the inner terms.(41t2+5)(t+3)=41t2⋅t+41t2⋅3+5⋅t+⋯
- Now, multiply the last terms.(41t2+5)(t+3)=41t2⋅t+41t2⋅3+5⋅t+5⋅3
- Finally, simplify the resulting expression by performing the required multiplications.(41t2+5)(t+3)=41t3+43t2+5t+15
V(t)=(-91t2+2t+2)(41t3+43t2+5t+15)
▼
Multiply (-91t2+2t+2) by (41t3+43t2+5t+15)
Distr
Distribute -91t2+2t+2
V(t)=(-91t2+2t+2)41t3+(-91t2+2t+2)43t2+(-91t2+2t+2)5t+(-91t2+2t+2)15
Distr
Distribute 41t3,43t2,5t,&15
V(t)=(-91t2⋅41t3+2t⋅41t3+2⋅41t3)+(-91t2⋅43t2+2t⋅43t2+2⋅43t2)+(-91t2⋅5t+2t⋅5t+2⋅5t)+(-91t2⋅15+2t⋅15+2⋅15)
CommutativePropMult
Commutative Property of Multiplication
V(t)=(-91⋅41t2⋅t3+2⋅41t⋅t3+2⋅41t3)+(-91⋅43t2⋅t2+2⋅43t⋅t2+2⋅43t2)+(-91⋅5t2⋅t+2⋅5t⋅t+2⋅5t)+(-91⋅15t2+2⋅15t+2⋅15)
▼
Simplify right-hand side
MultPow
am⋅an=am+n
V(t)=(-91⋅41t5+2⋅41t4+2⋅41t3)+(-91⋅43t4+2⋅43t3+2⋅43t2)+(-91⋅5t3+2⋅5t2+2⋅5t)+(-91⋅15t2+2⋅15t+2⋅15)
MultFrac
Multiply fractions
V(t)=(-361t5+42t4+42t3)+(-363t4+46t3+46t2)+(-95t3+2⋅5t2+2⋅5t)+(-915t2+2⋅15t+2⋅15)
SimpQuotProd
Simplify quotient and product
V(t)=(-361t5+21t4+21t3)+(-121t4+23t3+23t2)+(-95t3+10t2+10t)+(-35t2+30t+30)
CommutativePropAdd
Commutative Property of Addition
V(t)=-361t5+(21t4−121t4)+(21t3−95t3+23t3)+(23t2−35t2+10t2)+(10t+30t)+30
AddSubTerms
Add and subtract terms
V(t)=-361t5+125t4+913t3+659t2+40t+30
b The polynomial V(t) found in Part A gives the volume of the container after passing t hours underwater. It is given that Izabella's nephew put the container underwater at 10:00 AM and took it out at 4:00 PM. Therefore, the container was underwater for 6 hours.
6hours10:00 AM⟶4:00 PM
Consequently, to find the current volume of the container, evaluate V(t) at t=6.
V(t)=-361t5+125t4+913t3+659t2+40t+30
▼
Substitute 6 for t and evaluate
Substitute
t=6
V(6)=-361(6)5+125(6)4+913(6)3+659(6)2+40(6)+30
CalcPow
Calculate power
V(6)=-361⋅7776+125⋅1296+913⋅216+659⋅36+40(6)+30
MoveRightFacToNum
ca⋅b=ca⋅b
V(6)=-367776+125⋅1296+913⋅216+659⋅36+40(6)+30
Multiply
Multiply
V(6)=-367776+126480+92808+62124+240+30
CalcQuot
Calculate quotient
V(6)=-216+540+312+354+240+30
AddSubTerms
Add and subtract terms
V(6)=1260
Discussion
Analyzing the Product of Two Polynomials
In all the examples seen throughout this lesson, the product of two or more polynomials has resulted in a new polynomial. This is not a coincidence. In fact, the following property guarantees that multiplying polynomials always produces a polynomial.
Rule
Closure Property of Polynomial Multiplication
Given two polynomials P(x) and Q(x), the product P(x)⋅Q(x) is always a polynomial.
Multiplying two polynomials produces a new polynomial.
In other words, the polynomials are closed under multiplication.
Proof
Consider two arbitrary polynomials P(x) and Q(x) written in standard form.
Since P(x) and Q(x) are polynomials, all the exponents are whole numbers. Furthermore, because the whole numbers are closed under addition, the exponents of the resulting expression are whole numbers. Then, the new expression can be rewritten as follows.
P(x)Q(x)=anxd1+⋯+a1x+a0=bnxd2+⋯+b1x+b0
These two polynomials can be multiplied by using the Distributive Property. The Product of Powers Property can also be applied to simplify the resulting expression.
P(x)Q(x)=ckxD+⋯+c1x+c0
Consequently, the new expression is a polynomial. Therefore, the product of two polynomials produces a polynomial, which proves that the polynomials are closed under multiplication.
Pop Quiz
Practicing Multiplication of Polynomials
Given two polynomials P(x) and Q(x), compute their product and find the required information.
Closure
Computing a Polynomials Product
Before finishing this lesson, take another look at the challenge presented at the beginning. There it was given the following pair of polynomials.
Having both polynomials written in standard form, the required product can be found by using the Distributive Property.
As expected, the resulting expression is a polynomial. The degree of the resulting polynomial is 5, its leading coefficient is 8, and its constant term is -40.
P(x)Q(x)=2x3−x+5=4(x+2)(x−1)
In the challenge, it was asked whether it was possible to find the product of P(x) and Q(x). Now, with all the knowledge learned, it can be said that it is possible to multiply these two polynomials, and the resulting expression will also be a polynomial. To compute P(x)⋅Q(x), first, rewrite Q(x) in standard form. To do so, the FOIL Method can be applied.
Q(x)=4(x+2)(x−1)
MultII
Multiply (x+2) by (x−1)
Q(x)=4(x(x)+x(-1)+2x+2(-1))
▼
Simplify right-hand side
MultPow
am⋅an=am+n
Q(x)=4(x2+x(-1)+2x+2(-1))
MultPosNeg
a(-b)=-a⋅b
Q(x)=4(x2−x⋅1+2x−2⋅1)
MultByOne
a⋅1=a
Q(x)=4(x2−x+2x−2)
AddTerms
Add terms
Q(x)=4(x2+x−2)
Distr
Distribute 4
Q(x)=4x2+4x−8
P(x)⋅Q(x)
SubstituteExpressions
Substitute expressions
(2x3−x+5)(4x2+4x−8)
(2x3⋅4x2−x⋅4x2+5⋅4x2)+(2x3⋅4x−x⋅4x+5⋅4x)+(2x3(-8)−x(-8)+5(-8))
▼
Simplify
MultPow
am⋅an=am+n
(2⋅4x5−4x3+5⋅4x2)+(2⋅4x4−4x2+5⋅4x)+(2x3(-8)−x(-8)+5(-8))
Multiply
Multiply
(8x5−4x3+20x2)+(8x4−4x2+20x)+(-16x3+8x−40)
CommutativePropAdd
Commutative Property of Addition
8x5+8x4+(-4x3−16x3)+(20x2−4x2)+(20x+8x)−40
AddSubTerms
Add and subtract terms
8x5+8x4−20x3+16x2+28x−40
Multiplying Polynomials
Exercise 3.1
Find the product of the following polynomials. Write the answer in standard form.
P(x)Q(x)=x2−2x+1=2x3+3x2−x−2
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Let's find the required product using the Distributive Property.
Next, we continue simplifying the product by applying the Product of Powers Property and combining like terms.
One of the given polynomials has more than 3 terms. In this case, it could be convenient to use the Box Method to multiply them. Let's start by counting the number of terms.
ccc
Polynomial & & Number of Terms
P(x) & & 3
Q(x) & & 4
We will need to draw a table with three rows and four columns. We will place the terms of P(x) to the left of the table and the term of Q(x) above the table.
Next, we 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^2 and 2x^3. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.
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Find the product of the following polynomials. Write the answer in standard form.
P(x)Q(x)=x4+3x2−x+1=2x3+x2+x−3
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<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 mathdefault\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathdefault\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["2x^7 + x^6 + 7x^5 - 2x^4 + 4x^3 - 9x^2 + 4x - 3"]}}
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The given polynomials have more than 3 terms. In this case, it could be convenient to use the Box Method to multiply them. Then, let's start by counting the number of terms. ccc Polynomial & & Number of Terms P(x) & & 4 Q(x) & & 4 We will need to draw a table with four rows and four columns. We will place the terms of P(x) to the left of the table and the terms of Q(x) above the table.
Next, we 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 2x^4 and x^4. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.
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Find the product of the following three polynomials. Write the answer in standard form.
P(x)Q(x)R(x)=x+3=3x2−2x+4=x2+2x−5
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<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 mathdefault\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathdefault\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["3x^5 + 13x^4 - 3x^3 - 27x^2 + 34x - 60"]}}
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To make the computations easier, let's first find the product of P(x) and Q(x) and then, we will multiply the resulting polynomial by R(x). P(x)Q(x) = (x + 3)(3x^2 - 2x + 4) These two polynomials can be multiplied by using the Distributive Property.
As we can see, the resulting polynomial has 4 terms and, since R(x) has 3 terms, it could be convenient to multiply them using the Box Method. Let's make a 4 by 3 table and write the terms of P(x)Q(x) to the left of the table and the terms of R(x) above the table.
Next, we 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 3x^3 and x^2. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.
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Multiplying Polynomials
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