1. 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.

2. 7-8 Multiplying Polynomials Multiply. Additional Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) (3mn 2 )(9m 2 n) 27m 3 n 3 Multiply. Group factors with like bases together. Multiply. (6 3)(y 3 y 5 )   (3 9)(m m 2 )(n 2  n)  

3. 7-8 Multiplying Polynomials Multiply. Additional Example 1C: Multiplying Monomials Group factors with like bases together. Multiply.       222 1 12 4 tstt s s     g g g g g      22 1 4 t ststs   2

4. 7-8 Multiplying Polynomials When multiplying powers with the same base, keep the base and add the exponents. x 2  x 3 = x 2+3 = x 5 Remember!

5. 7-8 Multiplying Polynomials Check It Out! Example 1 Multiply. a. (3x 3 )(6x 2 ) (3x 3 )(6x 2 ) (3 6)(x 3 x 2 )   18x 5 Group factors with like bases together. Multiply. Group factors with like bases together. Multiply. b. (2r 2 t)(5t 3 ) (2r 2 t)(5t 3 ) (2 5)(r 2 )(t 3 t)   10r 2 t 4

6. 7-8 Multiplying Polynomials Check It Out! Example 1 Multiply. Group factors with like bases together. Multiply. c.c.     4522 1 12 3 xzyzx y    3    1 (12x 3 z 2 )(y 4 z 5 ) 3 x2yx2y     gggg    32245 1 12 z 3 zx y    755 4xyz

7. 7-8 Multiplying Polynomials Check up: p. 451 #’s 3,4,6

8. 7-8 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property.

9. 7-8 Multiplying Polynomials Multiply. Additional Example 2A: Multiplying a Polynomial by a Monomial 4(3x 2 + 4x – 8) (4)3x 2 + (4)4x – (4)8 12x 2 + 16x – 32 Distribute 4. Multiply.

10. 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) 12p 2 q – 6pq 2 Distribute 6pq. Group like bases together. Multiply.

11. 7-8 Multiplying Polynomials Multiply. Additional Example 2C: Multiplying a Polynomial by a Monomial Group like bases together. Multiply. 1 2 x2yx2y(6xy + 8x 2 y 2 ) x y   2 2 6 1 2 yx 8 2       2 2 1 6 2 8 2 2    1 2       x 2 x         1 6 2 y x 2 y y 2 8          1 2 3x 3 y 2 + 4x 4 y 3 Distribute. 2 1 x y 2

12. 7-8 Multiplying Polynomials Check It Out! Example 2 Multiply. a. 2(4x 2 + x + 3) 2(4x 2 + x + 3) 2(4x 2 ) + 2(x) + 2(3) 8x 2 + 2x + 6 Distribute 2. Multiply.

13. 7-8 Multiplying Polynomials Check It Out! Example 2 Multiply. 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. Multiply.

14. 7-8 Multiplying Polynomials Check It Out! Example 2 Multiply. c. 5r 2 s 2 (r – 3s) 5r 2 s 2 (r – 3s) (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 Distribute 5r 2 s 2. Group like bases together. Multiply.

15. 7-8 Multiplying Polynomials Check up: p. 451 #’s 10,12

16. 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) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6

17. 7-8 Multiplying Polynomials Another 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) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 F OIL 1. Multiply the First terms. (x + 3)(x + 2) x x = x 2 

18. 7-8 Multiplying Polynomials Multiply. Additional Example 3A: Multiplying Binomials (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 s and 4. Distribute s and 4 again. Multiply. Combine like terms.

19. 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) x 2 – 4x – 4x + 16 x 2 – 8x + 16 Write as a product of two binomials. Use the FOIL method. Multiply. Combine like terms.

20. 7-8 Multiplying Polynomials Additional Example 3C: Multiplying Binomials Multiply. (8m 2 – n)(m 2 – 3n) 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 Use the FOIL method. Multiply. Combine like terms.

21. 7-8 Multiplying Polynomials In the expression (x + 5) 2, the base is (x + 5). (x + 5) 2 = (x + 5)(x + 5) Helpful Hint

22. 7-8 Multiplying Polynomials Check It Out! Example 3a Multiply. (a + 3)(a – 4) a(a – 4)+ 3(a – 4) a(a) + a(–4) + 3(a) + 3(–4) a 2 – a – 12 a 2 – 4a + 3a – 12 Distribute a and 3. Distribute a and 3 again. Multiply. Combine like terms.

23. 7-8 Multiplying Polynomials Check It Out! Example 3b Multiply. (x – 3) 2 (x – 3)(x – 3) x(x) + x(–3) – 3(x) – 3(–3) x 2 – 3x – 3x + 9 x 2 – 6x + 9 Write as a product of two binomials. Use the FOIL method. Multiply. Combine like terms.

24. 7-8 Multiplying Polynomials Check It Out! Example 3c Multiply. (2a – b 2 )(a + 4b 2 ) 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 Use the FOIL method. Multiply. Combine like terms.

25. 7-8 Multiplying Polynomials Check up: p. 451 #’s 14, 17

26. 7-8 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – 6): (5x + 3)(2x 2 + 10x – 6) = 5x(2x 2 + 10x – 6) + 3(2x 2 + 10x – 6) = 5x(2x 2 + 10x – 6) + 3(2x 2 + 10x – 6) = 5x(2x 2 ) + 5x(10x) + 5x( – 6) + 3(2x 2 ) + 3(10x) + 3( – 6) = 10x 3 + 50x 2 – 30x + 6x 2 + 30x – 18 = 10x 3 + 56x 2 – 18

27. 7-8 Multiplying Polynomials 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 2 + 10x – 6) and width (5x + 3): 2x22x2 +10x –6–6 10x 3 50x 2 – 30x 30x6x26x2 – 18 5x5x +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. 10x 3 + 6x 2 + 50x 2 + 30x – 30x – 18 10x 3 + 56x 2 – 18

28. 7-8 Multiplying Polynomials 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 2 + 10x – 6 5x + 3  6x 2 + 30x – 18 Multiply each term in the top polynomial by 3. Multiply each term in the top polynomial by 5x, and align like terms. + 10x 3 + 50x 2 – 30x 10x 3 + 56x 2 + 0x – 18 10x 3 + 56x 2 – 18 Combine like terms by adding vertically. Simplify.

29. 7-8 Multiplying Polynomials Multiply. Additional Example 4A: Multiplying Polynomials (x – 5)(x 2 + 4x – 6) x(x 2 + 4x – 6) – 5(x 2 + 4x – 6) x(x 2 ) + x(4x) + x(–6) – 5(x 2 ) – 5(4x) – 5(–6) x 3 + 4x 2 – 6x – 5x 2 – 20x + 30 x 3 – x 2 – 26x + 30 Distribute x and –5. Distribute x and −5 again. Simplify. Combine like terms.

30. 7-8 Multiplying Polynomials Multiply. Additional Example 4B: Multiplying Polynomials (2x – 5)(–4x 2 – 10x + 3) –4x 2 – 10x + 3 2x – 5 × 20x 2 + 50x – 15 + –8x 3 – 20x 2 + 6x –8x 3 + 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.

31. 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) (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. Multiply. Combine like terms.

32. 7-8 Multiplying Polynomials Additional Example 4C Continued Multiply. (x + 3) 3 x 3 + 6x 2 + 9x + 3x 2 + 18x + 27 x 3 + 9x 2 + 27x + 27 x(x 2 ) + x(6x) + x(9) + 3(x 2 ) + 3(6x) + 3(9) x(x 2 + 6x + 9) + 3(x 2 + 6x + 9) Use the Commutative Property of Multiplication. Distribute the x and 3. Distribute the x and 3 again. (x + 3)(x 2 + 6x + 9) Combine like terms.

33. 7-8 Multiplying Polynomials Additional Example 4D: Multiplying Polynomials Multiply. (3x + 1)(x 3 + 4x 2 – 7) x3x3 4x24x2 –7–7 3x43x4 12x 3 – 21x 4x24x2 x3x3 –7–7 3x3x +1 3x 4 + 12x 3 + x 3 + 4x 2 – 21x – 7 Write the product of the monomials in each row and column. Add all terms inside the rectangle. 3x 4 + 13x 3 + 4x 2 – 21x – 7 Combine like terms.

34. 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 

35. 7-8 Multiplying Polynomials Check It Out! Example 4a Multiply. (x + 3)(x 2 – 4x + 6) x(x 2 – 4x + 6) + 3(x 2 – 4x + 6) Distribute x and 3. Distribute x and 3 again. x(x 2 ) + x(–4x) + x(6) + 3(x 2 ) + 3(–4x) + 3(6) x 3 – 4x 2 + 6x + 3x 2 – 12x + 18 x 3 – x 2 – 6x + 18 Simplify. Combine like terms.

36. 7-8 Multiplying Polynomials Check It Out! Example 4b Multiply. (3x + 2)(x 2 – 2x + 5) x 2 – 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. 2x 2 – 4x + 10 + 3x 3 – 6x 2 + 15x 3x 3 – 4x 2 + 11x + 10 Combine like terms by adding vertically.

37. 7-8 Multiplying Polynomials Check up: p. 451 #’s 22,24

38. 7-8 Multiplying Polynomials Additional 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 = h 2 – 3h + 4h – 12 Combine like terms. A = h 2 + h – 12 The area is represented by h 2 + h – 12.

39. 7-8 Multiplying Polynomials Additional 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 = h 2 + h – 12 A = 5 2 + 5 – 12 A = 25 + 5 – 12 A = 18 Use the polynomial from part A. Substitute 5 for h. Simplify. Combine terms. The area is 18 square feet.

40. 7-8 Multiplying Polynomials 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. Write the formula for the area of a rectangle. Substitute x – 4 for l and x for w. A = l w   A = (x – 4)x Multiply. A = x 2 – 4x The area is represented by x 2 – 4x.

41. 7-8 Multiplying Polynomials Check 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 = x 2 – 4x A = 36 – 24 A = 12 Use the polynomial from part a. Substitute 6 for x. Simplify. Combine terms. The area is 12 square meters. A = 6 2 – 4(6)

42. 7-8 Multiplying Polynomials Check up: p. 451 # 25