7-8 Multiplying Polynomials
To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
7-8 Multiplying Polynomials
Multiply.
Additional Example 1: Multiplying Monomials
A. (6y3)(3y5) (6y3)(3y5)
18y8
Group factors with like bases together.
B. (3mn2) (9m2n) (3mn2)(9m2n)
27m3n3
Multiply.
Group factors with like bases together.
Multiply.
(6 3)(y3 y5)
(3 9)(m m2)(n2 n)
7-8 Multiplying Polynomials
Multiply.
Additional Example 1C: Multiplying Monomials
2 2 21 124 s t st st
4 53s t
Group factors with like bases together.
Multiply.
22 21 124 ts tt s s
2 21 124 t s ts ts
2
7-8 Multiplying Polynomials
When multiplying powers with the same base, keep the base and add the exponents. x2 x3 = x2+3 = x5
Remember!
7-8 Multiplying PolynomialsCheck It Out! Example 1
Multiply.a. (3x3)(6x2)
(3x3)(6x2) (3 6)(x3 x2)
18x5
Group factors with like bases together.
Multiply.
Group factors with like bases together.
Multiply.
b. (2r2t)(5t3)
(2r2t)(5t3) (2 5)(r2)(t3 t)
10r2t4
7-8 Multiplying PolynomialsCheck It Out! Example 1
Multiply.
Group factors with like bases together.
Multiply.
c.
4 52 21 123 x zy zx y
3
1 (12x3z2)(y4z5)3 x2y
3 22 4 51 12 z3 zx x y y7554x y z
7-8 Multiplying Polynomials
Check up: p. 451 #’s 3,4,6
7-8 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property.
7-8 Multiplying Polynomials
Multiply.
Additional Example 2A: Multiplying a Polynomial by a Monomial
4(3x2 + 4x – 8)
4(3x2 + 4x – 8)
(4)3x2 + (4)4x – (4)8
12x2 + 16x – 32
Distribute 4.
Multiply.
7-8 Multiplying Polynomials
6pq(2p – q)
(6pq)(2p – q)
Multiply.
Additional Example 2B: Multiplying a Polynomial by a Monomial
(6pq)2p + (6pq)(–q) (6 2)(p p)(q) + (–1)(6)(p)(q q)
12p2q – 6pq2
Distribute 6pq.
Group like bases together.
Multiply.
7-8 Multiplying Polynomials
Multiply.
Additional Example 2C: Multiplying a Polynomial by a Monomial
Group like bases together.
Multiply.
12 x2y (6xy + 8x2y2)
x y 22 612 xyyx 8 2
x y x y
2 21 62 8xy x y 22
12
x2 • x
1
• 62 y • y x2 • x2 y • y2• 8
12
3x3y2 + 4x4y3
Distribute .21 x y2
7-8 Multiplying PolynomialsCheck It Out! Example 2
Multiply.a. 2(4x2 + x + 3)
2(4x2 + x + 3)
2(4x2) + 2(x) + 2(3)
8x2 + 2x + 6
Distribute 2.
Multiply.
7-8 Multiplying PolynomialsCheck It Out! Example 2
Multiply.b. 3ab(5a2 + b)
3ab(5a2 + b)
(3ab)(5a2) + (3ab)(b)
(3 5)(a a2)(b) + (3)(a)(b b)
15a3b + 3ab2
Distribute 3ab.
Group like bases together.
Multiply.
7-8 Multiplying PolynomialsCheck It Out! Example 2
Multiply.c. 5r2s2(r – 3s)
5r2s2(r – 3s)
(5r2s2)(r) – (5r2s2)(3s)
(5)(r2 r)(s2) – (5 3)(r2)(s2 s)
5r3s2 – 15r2s3
Distribute 5r2s2.
Group like bases together.
Multiply.
7-8 Multiplying Polynomials
Check up: p. 451 #’s 10,12
7-8 Multiplying Polynomials
To multiply a binomial by a binomial, you can apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.
Distribute x and 3 again.
Multiply.
Combine like terms.
= x(x + 2) + 3(x + 2)
= x(x) + x(2) + 3(x) + 3(2)
= x2 + 2x + 3x + 6
= x2 + 5x + 6
7-8 Multiplying PolynomialsAnother method for multiplying binomials is called the FOIL method.
4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6
3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x
2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x
F
O
I
L
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F O I L
1. Multiply the First terms. (x + 3)(x + 2) x x = x2
7-8 Multiplying Polynomials
Multiply.
Additional Example 3A: Multiplying Binomials
(s + 4)(s – 2)(s + 4)(s – 2)s(s – 2) + 4(s – 2)s(s) + s(–2) + 4(s) + 4(–2)
s2 – 2s + 4s – 8s2 + 2s – 8
Distribute s and 4.
Distribute s and 4 again.
Multiply.
Combine like terms.
7-8 Multiplying Polynomials
Multiply.
Additional Example 3B: Multiplying Binomials
(x – 4)2
(x – 4)(x – 4)
x(x) + x(–4) – 4(x) – 4(–4)
x2 – 4x – 4x + 16
x2 – 8x + 16
Write as a product of two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
7-8 Multiplying PolynomialsAdditional Example 3C: Multiplying Binomials
Multiply.(8m2 – n)(m2 – 3n)
8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)
8m4 – 24m2n – m2n + 3n2
8m4 – 25m2n + 3n2
Use the FOIL method.
Multiply.
Combine like terms.
7-8 Multiplying Polynomials
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint
7-8 Multiplying PolynomialsCheck It Out! Example 3a
Multiply. (a + 3)(a – 4)
(a + 3)(a – 4) a(a – 4)+ 3(a – 4) a(a) + a(–4) + 3(a) + 3(–4)
a2 – a – 12 a2 – 4a + 3a – 12
Distribute a and 3.
Distribute a and 3 again.
Multiply.
Combine like terms.
7-8 Multiplying PolynomialsCheck It Out! Example 3b
Multiply.
(x – 3)2
(x – 3)(x – 3)
x(x) + x(–3) – 3(x) – 3(–3)
x2 – 3x – 3x + 9
x2 – 6x + 9
Write as a product of two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
7-8 Multiplying PolynomialsCheck It Out! Example 3c
Multiply.(2a – b2)(a + 4b2)
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4
2a2 + 7ab2 – 4b4
Use the FOIL method.
Multiply.
Combine like terms.
7-8 Multiplying Polynomials
Check up: p. 451 #’s 14, 17
7-8 Multiplying PolynomialsTo multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18
7-8 Multiplying PolynomialsYou can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3):
2x2 +10x –610x3 50x2 –30x
30x6x2 –185x
+3
Write the product of the monomials in each row and column:
To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x3 + 6x2 + 50x2 + 30x – 30x – 18
10x3 + 56x2 – 18
7-8 Multiplying PolynomialsAnother method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers.
2x2 + 10x – 6
5x + 36x2 + 30x – 18
Multiply each term in the top polynomial by 3.
Multiply each term in the top polynomial by 5x, and align like terms.+ 10x3 + 50x2 – 30x
10x3 + 56x2 + 0x – 18
10x3 + 56x2 – 18
Combine like terms by adding vertically.
Simplify.
7-8 Multiplying Polynomials
Multiply.
Additional Example 4A: Multiplying Polynomials
(x – 5)(x2 + 4x – 6)(x – 5 )(x2 + 4x – 6)x(x2 + 4x – 6) – 5(x2 + 4x – 6)
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 6x – 5x2 – 20x + 30
x3 – x2 – 26x + 30
Distribute x and –5.
Distribute x and −5 again.
Simplify.
Combine like terms.
7-8 Multiplying Polynomials
Multiply.
Additional Example 4B: Multiplying Polynomials
(2x – 5)(–4x2 – 10x + 3)
(2x – 5)(–4x2 – 10x + 3)
–4x2 – 10x + 32x – 5×
20x2 + 50x – 15+ –8x3 – 20x2 + 6x
–8x3 + 56x – 15
Multiply each term in the top polynomial by –5.
Multiply each term in the top polynomial by 2x, and align like terms.
Combine like terms by adding vertically.
7-8 Multiplying Polynomials
Multiply.
Additional Example 4C: Multiplying Polynomials
(x + 3)3
[(x + 3)(x + 3)](x + 3)
[x(x) + x(3) + 3(x) +3(3)](x + 3)
(x2 + 3x + 3x + 9)(x + 3)
(x2 + 6x + 9)(x + 3)
Write as the product of three binomials.
Use the FOIL method on the first two factors.
Multiply.
Combine like terms.
7-8 Multiplying PolynomialsAdditional Example 4C Continued
Multiply.
(x + 3)3
x3 + 6x2 + 9x + 3x2 + 18x + 27
x3 + 9x2 + 27x + 27
x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)
x(x2 + 6x + 9) + 3(x2 + 6x + 9)
Use the Commutative Property of Multiplication.
Distribute the x and 3.
Distribute the x and 3 again.
(x + 3)(x2 + 6x + 9)
Combine like terms.
7-8 Multiplying PolynomialsAdditional Example 4D: Multiplying Polynomials
Multiply.(3x + 1)(x3 + 4x2 – 7)
x3 4x2 –73x4 12x3 –21x
4x2x3 –73x+1
3x4 + 12x3 + x3 + 4x2 – 21x – 7
Write the product of the monomials in each row and column.
Add all terms inside the rectangle.
3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.
7-8 Multiplying Polynomials
A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying.
Helpful Hint
7-8 Multiplying PolynomialsCheck It Out! Example 4a
Multiply.(x + 3)(x2 – 4x + 6)
(x + 3)(x2 – 4x + 6)x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute x and 3.
Distribute x and 3 again.
x(x2) + x(–4x) + x(6) + 3(x2) + 3(–4x) + 3(6)
x3 – 4x2 + 6x + 3x2 – 12x + 18
x3 – x2 – 6x + 18
Simplify.
Combine like terms.
7-8 Multiplying PolynomialsCheck It Out! Example 4b
Multiply.(3x + 2)(x2 – 2x + 5)
(3x + 2)(x2 – 2x + 5)
x2 – 2x + 5 3x + 2
Multiply each term in the top polynomial by 2.
Multiply each term in the top polynomial by 3x, and align like terms.2x2 – 4x + 10
+ 3x3 – 6x2 + 15x3x3 – 4x2 + 11x + 10
Combine like terms by adding vertically.
7-8 Multiplying Polynomials
Check up: p. 451 #’s 22,24
7-8 Multiplying PolynomialsAdditional Example 5: Application
The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.A. Write a polynomial that represents the
area of the base of the prism.Write the formula for the
area of a rectangle. Substitute h – 3 for w
and h + 4 for l.
A = l w
A = l w A = (h + 4)(h – 3)
Multiply.A = h2 – 3h + 4h – 12Combine like terms.A = h2 + h – 12
The area is represented by h2 + h – 12.
7-8 Multiplying PolynomialsAdditional Example 5: Application
The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.B. Find the area of the base when the
height is 5 ft.A = h2 + h – 12
A = 52 + 5 – 12A = 25 + 5 – 12A = 18
Use the polynomial from part A.
Substitute 5 for h.
Simplify.
Combine terms.The area is 18 square feet.
7-8 Multiplying PolynomialsCheck It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.a. Write a polynomial that represents the area
of the rectangle.
Write the formula for the area of a rectangle.
Substitute x – 4 for l and x for w.
A = l w
A = l w
A = (x – 4)xMultiply.A = x2 – 4x
The area is represented by x2 – 4x.
7-8 Multiplying PolynomialsCheck It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.b. Find the area of a rectangle when the width
is 6 meters.
A = x2 – 4x
A = 36 – 24A = 12
Use the polynomial from part a.
Substitute 6 for x.
Simplify.
Combine terms.
The area is 12 square meters.
A = 62 – 4(6)
7-8 Multiplying Polynomials
Check up: p. 451 # 25