CHAPTER 6 6-5 Multiplying Polynomials

OBJECTIVES Multiply polynomials.

MULTIPLYING POLYNOMIALS To multiply monomials and

polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

EXAMPLE 1: MULTIPLYING MONOMIALS

Multiply. A. (6y3)(3y5) Solution: (6y3)(3y5) Group factors with like bases

together. (6 *3)(y3 * y5) Multiply. 18y8

EXAMPLE#1 B. (3mn2) (9m2n) Solution: (3mn2)(9m2n) Group factors with like

bases together. (3 *9)(m * m2)(n2 n) Multiply

27m3n3

EXAMPLE#1 Multiply

Solution:

14 s2 t2 (st) (-12 s t2)

4 53s t

CHECK IT OUT! a. (3x3)(6x2) Solution: 18x5

b. (2r2t)(5t3) Solution: 10r2t4

MULTIPLYING POLYNOMIALS BY MONOMIALS

To multiply a polynomial by a monomial, use the Distributive Property.

EXAMPLE 2A: MULTIPLYING A POLYNOMIAL BY A MONOMIAL

Multiply. 4(3x2 + 4x – 8) sol:

4(3x2 + 4x – 8) (4)3x2 +(4)4x – (4)8 multiply

12x2 + 16x – 32

Distribute 4.

EXAMPLE 2B: MULTIPLYING A POLYNOMIAL BY A MONOMIAL

Multiply. 6pq(2p – q) Sol:

(6pq)(2p – q) (6pq)2p + (6pq)(–q) 6 2)(p p)(q) + (–1)(6)(p)(q q) 12p2q – 6pq2

Distribute 6pq.

CHECK IT OUT! Multiply a. 2(4x2 + x + 3) Sol: 8x2 + 2x + 6 b. 3ab(5a2 + b) Sol:15a3b + 3ab2

MULTIPLYING BINOMIALS BY BINOMIALSTo 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)

x(x + 2) + 3(x + 2) x(x) + x(2) + 3(x) + 3(2) Multiply x2 + 2x + 3x + 6 combine x2 + 5x + 6

Distribute.

Distribute again.

MULTIPLYING BINOMIAL BY BINOMIAL Another method for multiplying

binomials is called : FOIL Method 1. Multiply the First terms. (x + 3)(x + 2) x x = x2

2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x

3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x

4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6

(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6

F O I L

EXAMPLE Multiply. (s + 4)(s – 2) Solution: s2 + 2s – 8

EXAMPLE 3B: MULTIPLYING BINOMIALS

Multiply (x – 4)2

Solution: x2 – 8x + 16

EXAMPLE 3C: MULTIPLYING BINOMIALS

Multiply (8m2 – n)(m2 – 3n) Solution:8m4 – 25m2n + 3n2

CHECK IT OUT! EXAMPLE 3A

Multiply (a + 3)(a – 4) Solution: a2 – a – 12

MULTIPLYING POLYNOMIALS To multiply polynomials with more than

two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):

SOLUTION (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

BOX METHOD You 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:

BOX METHOD 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

EXAMPLE 4A: MULTIPLYING POLYNOMIALS

Multiply

S0l:(x – 5)(x2 + 4x – 6)

x3 – x2 – 26x + 30

EXAMPLE 4B: MULTIPLYING POLYNOMIALS

Multiply (2x – 5)(–4x2 – 10x + 3) Sol: –8x3 + 56x – 15

CHECK IT OUT!!! Multiply (3x + 1)(x3 + 4x2 – 7)

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.

Sol: The area is represented by h2 + h – 12.

CONTINUE b. Find the area of the base when

the height is 5 ft. Sol: The area is 18 square feet.

STUDENT GUIDED PRACTICE DO Problems 1,2,4,13,16,23 and 25 in

your book page 427

HOMEWORK Do problems 27,30,39,42,45,54,57,60

and 62 in your book page 427