6-5 Multiplying Polynomials
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1 6-5 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 1
2 Warm Up Evaluate Simplify (5 3 ) y 5 y (x 2 ) 4 y 9 x (x 7) 4x+ 28
3 Objective Multiply polynomials.
4 To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
5 Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 B. (3mn 2 ) (9m 2 n) (3mn 2 )(9m 2 n) (3 9)(m m 2 )(n 2 n) 27m 3 n 3 Group factors with like bases together. Group factors with like bases together.
6 Example 1C: Multiplying Monomials 1 s s2 t 2 (st) (-12 s t2 ) t s ( )( 2 t 12st ) ( 2 )( 2 2 s s s t t t ) Group factors with like bases together. 3st 4 5
7 Remember! When multiplying powers with the same base, keep the base and add the exponents. x 2 x 3 = x 2+3 = x 5
8 Check It Out!Example 1 a. (3x 3 )(6x 2 ) (3x 3 )(6x 2 ) (3 6)(x 3 x 2 ) 18x 5 b. (2r 2 t)(5t 3 ) (2r 2 t)(5t 3 ) (2 5)(r 2 )(t 3 t) 10r 2 t 4 Group factors with like bases together. Group factors with like bases together.
9 Check It Out! Example 1 Continued c x y x z y z ( )( 4 5 ) 1 x 3 y ( 3 )( xz y z ) z 3 ( 3)( )( 2 x 2 x y y 4 z 5 ) Group factors with like bases together xyz
10 To multiply a polynomial by a monomial, use the Distributive Property.
11 Example 2A: Multiplying a Polynomial by a Monomial 4(3x 2 + 4x 8) 4(3x 2 + 4x 8) Distribute 4. (4)3x 2 +(4)4x (4)8 12x x 32
12 Example 2B: Multiplying a Polynomial by a Monomial 6pq(2p q) (6pq)(2p q) Distribute 6pq. (6pq)2p + (6pq)( q) Group like bases together. (6 2)(p p)(q)+( 1)(6)(p)(q q) 12p 2 q 6pq 2
13 Example 2C: Multiplying a Polynomial by a Monomial 1 ( ) 2 2 x y xy x y ( 2 2 xy 6xy + 8x y ) Distribute. 2 1 x y x y ( ) ( 2 6xy + x y 8x y ) Group like bases together. ( x 2 1 x)( y y) + 8 ( x 2 x 2 )( y y 2 ) 2 3x 3 y 2 + 4x 4 y 3
14 Check It Out!Example 2 a. 2(4x 2 + x+ 3) 2(4x 2 + x+ 3) 2(4x 2 ) + 2(x) + 2(3) Distribute 2. 8x 2 + 2x+ 6
15 Check It Out!Example 2 b. 3ab(5a 2 + b) 3ab(5a 2 + b) (3ab)(5a 2 ) + (3ab)(b) (3 5)(a a 2 )(b) + (3)(a)(b b) 15a 3 b + 3ab 2 Distribute 3ab. Group like bases together.
16 Check It Out!Example 2 c. 5r 2 s 2 (r 3s) 5r 2 s 2 (r 3s) Distribute 5r 2 s 2. (5r 2 s 2 )(r) (5r 2 s 2 )(3s) (5)(r 2 r)(s 2 ) (5 3)(r 2 )(s 2 s) 5r 3 s 2 15r 2 s 3 Group like bases together.
17 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) = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) = x 2 + 2x + 3x + 6 Distribute. Distribute again. Combine like terms. = x 2 + 5x + 6
18 Another method for multiplying binomials is called the FOIL method. F 1. Multiply the Firstterms. (x + 3)(x + 2) x x = x 2 O 2. Multiply the Outerterms. (x + 3)(x + 2) x 2 = 2x I 3. Multiply the Innerterms. (x + 3)(x + 2) 3 x = 3x L 4. Multiply the Lastterms. (x + 3)(x + 2) 3 2 = 6 (x + 3)(x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 F O I L
19 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) s 2 2s+ 4s 8 s 2 + 2s 8 Distribute. Distribute again. Combine like terms.
20 Example 3B: Multiplying Binomials (x 4) 2 (x 4)(x 4) Write as a product of two binomials. Use the FOIL method. (x x)+ (x ( 4))+( 4 x)+( 4 ( 4)) x 2 4x 4x+16 x 2 8x+ 16 Combine like terms.
21 Example 3C: Multiplying Binomials (8m 2 n)(m 2 3n) Use the FOIL method. 8m 2 (m 2 )+ 8m 2 ( 3n) n(m 2 ) n( 3n) 8m 4 24m 2 n m 2 n+ 3n 2 8m 4 25m 2 n+ 3n 2 Combine like terms.
22 Helpful Hint In the expression (x+ 5) 2, the base is (x+ 5). (x + 5) 2 = (x+ 5)(x+ 5)
23 Check It Out!Example 3a (a+ 3)(a 4) (a+ 3)(a 4) a(a 4)+3(a 4) a(a) + a( 4) + 3(a)+ 3( 4) a 2 4a + 3a 12 a 2 a 12 Distribute. Distribute again. Combine like terms.
24 Check It Out!Example 3b (x 3) 2 (x 3)(x 3) Write as a product of two binomials. Use the FOIL method. (x x) +(x ( 3))+ ( 3 x)+( 3)( 3) x 2 3x 3x + 9 x 2 6x + 9 Combine like terms.
25 Check It Out!Example 3c (2a b 2 )(a + 4b 2 ) (2a b 2 )(a + 4b 2 ) Use the FOIL method. 2a(a)+ 2a(4b 2 ) b 2 (a)+ ( b 2 )(4b 2 ) 2a 2 + 8ab 2 ab 2 4b 4 2a 2 + 7ab 2 4b 4 Combine like terms.
26 To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x+ 3) by (2x x 6): (5x + 3)(2x x 6) = 5x(2x x 6) + 3(2x x 6) = 5x(2x x 6) + 3(2x x 6) = 5x(2x 2 ) + 5x(10x) + 5x( 6) + 3(2x 2 ) + 3(10x) + 3( 6) = 10x x 2 30x + 6x x 18 = 10x x 2 18
27 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 (2x x 6) and width (5x+ 3): 5x +3 2x 2 +10x 6 10x 3 50x 2 30x 6x 2 30x 18 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. 10x 3 + 6x x x 30x 18 10x x 2 18
28 Another 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. 2x x 6 5x + 3 6x x x x 2 30x 10x x 2 + 0x 18 10x x Multiply each term in the top polynomial by 3. Multiply each term in the top polynomial by 5x, and align like terms. Combine like terms by adding vertically. Simplify.
29 Example 4A: Multiplying Polynomials (x 5)(x 2 + 4x 6) (x 5 )(x 2 + 4x 6) x(x 2 + 4x 6) 5(x 2 + 4x 6) Distribute x. Distribute x again. x(x 2 ) + x(4x) + x( 6) 5(x 2 ) 5(4x) 5( 6) x 3 + 4x 2 5x 2 6x 20x+ 30 x 3 x 2 26x + 30 Simplify. Combine like terms.
30 Example 4B: Multiplying Polynomials (2x 5)( 4x 2 10x+ 3) (2x 5)( 4x 2 10x+ 3) 4x 2 10x+ 3 x 2x 5 20x x x 3 20x 2 + 6x 8x x 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.
31 Example 4C: Multiplying Polynomials (x+ 3) 3 [x x + x(3) + 3(x)+ (3)(3)] [x(x+3) + 3(x+3)](x+ 3) (x 2 + 3x + 3x+ 9)(x+ 3) (x 2 + 6x + 9)(x+ 3) Write as the product of three binomials. Use the FOIL method on the first two factors. Combine like terms.
32 Example 4C: Multiplying Polynomials Continued (x+ 3) 3 (x + 3)(x 2 + 6x+ 9) x(x 2 + 6x+ 9) + 3(x 2 + 6x+ 9) x(x 2 )+ x(6x)+ x(9) + 3(x 2 ) + 3(6x) + 3(9) x 3 + 6x 2 + 9x+ 3x x+ 27 Use the Commutative Property of Multiplication. Distribute. Distribute again. Combine like terms. x 3 + 9x x+ 27
33 Example 4D: Multiplying Polynomials (3x+ 1)(x 3 + 4x 2 7) 3x +1 x 3 4x 2 7 3x 4 12x 3 21x x 3 4x 2 7 Write the product of the monomials in each row and column. Add all terms inside the rectangle. 3x 4 12x 3 + x 3 4x 2 21x 7 3x 4 11x 3 4x 2 21x 7 Combine like terms.
34 Helpful Hint A polynomial with mterms multiplied by a polynomial with nterms has a product that, before simplifying has mnterms. In Example 4A, there are 2 3, or 6 terms before simplifying.
35 Check It Out!Example 4a (x+ 3)(x 2 4x+ 6) (x+ 3 )(x 2 4x+ 6) x(x 2 4x+ 6) + 3(x 2 4x+ 6) Distribute. Distribute again. x(x 2 ) + x( 4x) + x(6) +3(x 2 ) +3( 4x) +3(6) x 3 4x 2 + 3x 2 +6x 12x+ 18 x 3 x 2 6x + 18 Simplify. Combine like terms.
36 Check It Out!Example 4b (3x + 2)(x 2 2x+ 5) (3x + 2)(x 2 2x+ 5) x 2 2x+ 5 3x + 2 2x 2 4x x 3 6x x 3x 3 4x x + 10 Multiply each term in the top polynomial by 2. Multiply each term in the top polynomial by 3x, and align like terms. Combine like terms by adding vertically.
37 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. A = l w A = l w A =(h+ 4)(h 3) A = h 2 + 4h 3h 12 A = h 2 + h 12 Write the formula for the area of a rectangle. Substitute h 3 for w and h + 4 for l. Combine like terms. The area is represented by h 2 + h 12.
38 Example 5: Application Continued 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 = h 2 + h 12 A = h 2 + h 12 A = A = Write the formula for the area the base of the prism. Substitute 5 for h. Simplify. A = 18 Combine terms. The area is 18 square feet.
39 Check 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. A = l w A = l w A =x(x 4) A = x 2 4x The area is represented by x 2 4x. Write the formula for the area of a rectangle. Substitute x 4 for l and x for w.
40 Check It Out! Example 5 Continued 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 = x 2 4x Write the formula for the area of a A = x 2 rectangle whose length is 4 4x meters shorter than width. A = Substitute 6 for x. A = Simplify. A = 12 Combine terms. The area is 12 square meters.
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