Polynomials 6 5 Multiplying Polynomials Warm Up Lesson

Polynomials 6 -5 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Holt Mc. Dougal

6 -5 Multiplying Polynomials Warm Up Simplify and evaluate. 1. 7 m 2 + 3 m + 4 m 2 2. (r 2 + s 2) – (5 r 2 + 4 s 2) 3. (10 pq + 3 p) + (2 pq – 5 p + 6 pq) Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Essenstial Question How do you multiply polynomials? Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Remember! When multiplying powers with the same base, keep the base and add the exponents. x 2 x 3 = x 2+3 = x 5 Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example: Multiplying Monomials A. (6 y 3)(3 y 5) (6 3)(y 3 y 5) Group factors with like bases together. 18 y 8 Multiply. B. (3 mn 2) (9 m 2 n) (3 mn 2)(9 m 2 n) (3 9)(m m 2)(n 2 n) 27 m 3 n 3 Holt Mc. Dougal Algebra 1 Group factors with like bases together. Multiply.

6 -5 Multiplying Polynomials Example: Multiplying Monomials 1 2 2 s t 4 (st) (-12 s t 2) æ 1 2 2ö 2 s t t 12 t s ( ) s ç ÷ 4 è ø ( æ 1 ö 2 ç • − 12÷ s • s ø è 4 ( Holt Mc. Dougal Algebra 1 ) Group factors with like bases together. )(t • t ) 2 2 Multiply.

6 -5 Multiplying Polynomials I do…. Multiply. a. (3 x 3)(6 x 2) (3 6)(x 3 x 2) 18 x 5 Holt Mc. Dougal Algebra 1 Group factors with like bases together. Multiply.

6 -5 Multiplying Polynomials We do…. Multiply. b. (2 r 2 t)(5 t 3) Group factors with like bases together. (2 5)(r 2)(t 3 t) 10 r 2 t 4 Multiply. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials You do…. Multiply. æ 1 2 ö 3 2 4 5 x y 12 x z )( y z c. ç ÷( è 3 ø ) æ 1 2 ö 3 2 x y 12 x z ÷ ç ø è 3 )(y z ) ö 2 æ 1 3 • 12 x x • ÷ ç 3 ø è )( y • y )(z ( ( 4 x 5 y 5 z 7 Holt Mc. Dougal Algebra 1 4 5 4 Group factors with like bases together. 2 ) • z 5 Multiply.

6 -5 Multiplying Polynomials Distributive Property Of Multiplication Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example: Multiplying a Polynomial by a Monomial 4(3 x 2 + 4 x – 8) Distribute 4. (4)3 x 2 +(4)4 x – (4)8 Multiply. 12 x 2 + 16 x – 32 Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example Multiply. 6 pq(2 p – q) (6 pq)(2 p – q) Distribute 6 pq. (6 pq)2 p + (6 pq)(–q) (6 2)(p Group like bases together. p)(q) + (– 1)(6)(p)(q q) 12 p 2 q – 6 pq 2 Holt Mc. Dougal Algebra 1 Multiply.

6 -5 Multiplying Polynomials Example Multiply. 1 2 2 2 x y(6 xy + 8 x y ) 2 1 2 2 2 xy + 8 x y 6 2 ( 1 2 Distribute x y. 2 ) æ 1 2 ö 2 2 Group like bases + x y 6 xy x y 8 x y ( ) ÷ ç ø è 2 together. ö 2 æ 1 ç • 6 ÷ x • x ( y • y) + ç • 8÷ x • x 2 y • y 2 ø è 2 ( ( ) 3 x 3 y 2 + 4 x 4 y 3 Holt Mc. Dougal Algebra 1 ) ( )( Multiply. )

6 -5 Multiplying Polynomials FOIL method. F 1. Multiply the First terms. (x + 3)(x + 2) O 2. Multiply the Outer terms. (x + 3)(x + 2) I 3. Multiply the Inner terms. (x + 3)(x + 2) L 4. Multiply the Last terms. (x + 3)(x + 2) x x = x 2 x 2 = 2 x 3 x = 3 x 3 2 = 6 (x + 3)(x + 2) = x 2 + 2 x + 3 x + 6 = x 2 + 5 x + 6 F Holt Mc. Dougal Algebra 1 O I L

6 -5 Multiplying Polynomials Example: Multiplying Binomials (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute. s(s) + s(– 2) + 4(s) + 4(– 2) Distribute again. s 2 – 2 s + 4 s – 8 Multiply. s 2 + 2 s – 8 Combine like terms. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Helpful Hint In the expression (x + 5)2, the base is (x + 5)2 = (x + 5) Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example: Multiplying Binomials Multiply. (x – 4)2 (x – 4) Write as a product of two binomials. Use the FOIL method. (x x) + (x (– 4)) + (– 4 x) + (– 4)) x 2 – 4 x + 16 Multiply. x 2 – 8 x + 16 Combine like terms. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example Multiply. (x – 3)2 (x – 3) Write as a product of two binomials. Use the FOIL method. (x ● x) + (x (– 3)) + (– 3 x)+ (– 3) x 2 – 3 x + 9 Multiply. x 2 – 6 x + 9 Combine like terms. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Example: Multiplying Polynomials Multiply. (x – 5)(x 2 + 4 x – 6) (x – 5 )(x 2 + 4 x – 6) Distribute x. x(x 2 + 4 x – 6) – 5(x 2 + 4 x – 6) Distribute x again. x(x 2) + x(4 x) + x(– 6) – 5(x 2) – 5(4 x) – 5(– 6) x 3 + 4 x 2 – 5 x 2 – 6 x – 20 x + 30 Simplify. x 3 – x 2 – 26 x + 30 Combine like terms. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials 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. A = l w Substitute h – 3 for w A = (h + 4)(h – 3) and h + 4 for l. A = h 2 + 4 h – 3 h – 12 Multiply. A = l w A = h 2 + h – 12 Combine like terms. The area is represented by h 2 + h – 12. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials 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 Write the formula for the area the base of the prism. A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet. Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Lesson Quiz: Part I Multiply. 1. (6 s 2 t 2)(3 st) 2. 4 xy 2(x + y) 3. (x + 2)(x – 8) 4. (2 x – 7)(x 2 + 3 x – 4) 5. 6 mn(m 2 + 10 mn – 2) 6. (2 x – 5 y)(3 x + y) Holt Mc. Dougal Algebra 1

6 -5 Multiplying Polynomials Lesson Quiz: Part II 7. A triangle has a base that is 4 cm longer than its height. a. Write a polynomial that represents the area of the triangle. b. Find the area when the height is 8 cm. Holt Mc. Dougal Algebra 1