1. Copyright © Cengage Learning. All rights reserved. Polynomials 4

2. Copyright © Cengage Learning. All rights reserved. Section 4.6 Multiplying Polynomials

3. 3 Objectives Multiply two or more monomials. Multiply a polynomial by a monomial. Multiply a binomial by a binomial. Multiply a polynomial by a binomial. Solve an equation that simplifies to a linear equation. 1 1 2 2 3 3 4 4 5 5

4. 4 Objectives Solve an application involving multiplication of polynomials. 6 6

5. 5 Multiply two or more monomials 1.

6. 6 Multiply two or more monomials To multiply 4x 2 and –2x 3 Use the commutative and associative properties of multiplication to group the numerical factors together and the variable factors together. 4x 2 (–2x 3 ) = 4(–2)x 2 x 3 = –8x 5

7. 7 Multiply two or more monomials Multiplying Monomials To multiply two simplified monomials, multiply the numerical factors and then multiply the variable factors.

8. 8 Example Multiply. a. 3x 5 (2x 5 ) b. –2a 2 b 3 (5ab 2 ) Solution: a. 3x 5 (2x 5 ) = 3(2)x 5 x 5 = 6x 10 b. –2a 2 b 3 (5ab 2 ) = –2(5)a 2 ab 3 b 2 = –10a 3 b 5

9. 9 Multiply a polynomial by a monomial 2.

10. 10 Multiply a polynomial by a monomial To multiply 2x + 4 by 5x we proceed as follows: 5x(2x + 4) = 5x  2x + 5x  4 = 10x 2 + 20x

11. 11 Multiply a polynomial by a monomial Multiplying Polynomials by Monomials To multiply a polynomial with more than one term by a monomial, use the distributive property to remove parentheses and simplify.

12. 12 Example Multiply. a. 3a 2 (3a 2 – 5a) b. –2xz 2 (2x – 3z + 2z 2 ) Solution: a. 3a 2 (3a 2 – 5a) = 3a 2  3a 2 – 3a 2  5a = 9a 4 – 15a 3 Use the distributive property to remove Parentheses. Multiply.

13. 13 Example – Solution b. –2xz 2 (2x – 3z + 2z 2 ) = –2xz 2  2x + (–2xz 2 )  (–3z) + (–2xz 2 )  2z 2 = –4x 2 z 2 + 6xz 3 + (–4xz 4 ) = –4x 2 z 2 + 6xz 3 – 4xz 4 Recall that subtracting is equivalent to adding the opposite. cont’d Use the distributive Property to remove parentheses. Multiply.

14. 14 Multiply a binomial by a binomial 3.

15. 15 Multiply a binomial by a binomial To multiply two binomials, we must use the distributive property more than once. For example, to multiply (2a – 4) by (3a + 5), we proceed as follows. (2a – 4)(3a + 5) = (2a – 4)  3a + (2a – 4)  5 = 3a(2a – 4) + 5(2a – 4) = 3a  2a + 3a  (–4) + 5  2a + 5  (–4) Use the commutative property of multiplication. Use the distributive property to remove parentheses. Use the distributive property.

16. 16 Multiply a binomial by a binomial = 6a 2 – 12a + 10a – 20 = 6a 2 – 2a – 20 Multiplying Two Binomials To multiply two binomials, multiply each term of one binomial by each term of the other binomial and combine like terms. Do the multiplications. Combine like terms.

17. 17 Multiply a binomial by a binomial To multiply binomials, we can apply the distributive property using a mnemonic device, called the FOIL method. FOIL is an acronym for First terms, Outer terms, Inner terms, and Last terms. To use this method to multiply (2a – 4) by (3a + 5), we 1. multiply the First terms 2a and 3a to obtain 6a 2, 2. multiply the Outer terms 2a and 5 to obtain 10a, 3. multiply the Inner terms –4 and 3a to obtain –12a, and 4. multiply the Last terms –4 and 5 to obtain –20.

18. 18 Multiply a binomial by a binomial Then we simplify the resulting polynomial, if possible. = 2a(3a) + 2a(5) + (–4)(3a) + (–4)(5) = 6a 2 + 10a – 12a – 20 = 6a 2 – 2a – 20 Simplify. Combine like terms.

19. 19 Multiply a binomial by a binomial The products discussed in next example are called special products. These include squaring a binomial and multiplying conjugate binomials. Binomials that have the same terms, but with opposite signs between the terms, are called conjugate binomials.

20. 20 Example Find each product. a. (x + y) 2 = (x + y)(x + y) = x 2 + xy + xy + y 2 = x 2 + 2xy + y 2 The square of the sum of two quantities has three terms: the square of the first quantity, plus twice the product of the quantities, plus the square of the second quantity. Square the binomial. Distribute. Combine like terms.

21. 21 Example b. (x – y) 2 = (x – y)(x – y) = x 2 – xy – xy + y 2 = x 2 – 2xy + y 2 The square of the difference of two quantities has three terms: the square of the first quantity, minus twice the product of the quantities, plus the square of the second quantity. Square the binomial. Distribute. Combine like terms. cont’d

22. 22 Example c. (x + y)(x – y) = x 2 – xy + xy – y 2 = x 2 – y 2 The product of the sum and the difference of two quantities is a binomial. It is the product of the first quantities minus the product of the second quantities. Distribute. Combine like terms. cont’d

23. 23 Special Products (x + y) 2 = x 2 + 2xy + y 2 (x – y) 2 = x 2 – 2xy + y 2 (x + y)(x – y) = x 2 – y 2 Multiply a binomial by a binomial

24. 24 Multiply a polynomial by a binomial 4.

25. 25 Multiply a polynomial by a binomial We must use the distributive property more than once to multiply a polynomial by a binomial. For example, to multiply (3x 2 + 3x – 5) by (2x + 3), we proceed as follows: (2x + 3)(3x 2 + 3x – 5) = (2x + 3)3x 2 + (2x + 3)3x + (2x + 3)(–5) = 3x 2 (2x + 3) + 3x(2x + 3) – 5(2x + 3) = 6x 3 + 9x 2 + 6x 2 + 9x – 10x – 15 = 6x 3 + 15x 2 – x – 15

26. 26 Multiply a polynomial by a binomial Multiplying Polynomials To multiply one polynomial by another, multiply each term of one polynomial by each term of the other polynomial and combine like terms.

27. 27 Example a. Multiply:

28. 28 Example b. Multiply: cont’d

29. 29 Multiply a polynomial by a binomial Comment An expression can be simplified by combining its like terms. An equation (two quantities set equal) can be solved. Remember that Expressions are to be simplified. Expressions are to be solved.

30. 30 Solve an equation that simplifies to a linear equation 5.

31. 31 Solve an equation that simplifies to a linear equation To solve an equation such as (x + 2)(x + 3) = x(x + 7), we can use the distributive property to remove the parentheses on the left and the right sides and proceed as follows: (x + 2)(x + 3) = x(x + 7) x 2 + 3x + 2x + 6 = x 2 + 7x x 2 + 5x + 6 = x 2 + 7x 5x + 6 = 7x Combine like terms. Subtract x 2 from both sides. Use the distributive property to remove parentheses.

32. 32 Solve an equation that simplifies to a linear equation 6 = 2x 3 = x Check: (x + 2)(x + 3) = x(x + 7) (3 + 2)(3 + 3) ≟ 3(3 + 7) 5(6) ≟ 3(10) 30 = 30 Since the answer checks, the solution is 3. Replace x with 3. Do the additions within parentheses. Subtract 5x from both sides. Divide both sides by 2.

33. 33 Example Solve: (x + 5)(x + 4) = (x + 9)(x + 10) Solution: We remove parentheses on both sides of the equation and proceed as follows: (x + 5)(x + 4) = (x + 9)(x + 10) x 2 + 4x + 5x + 20 = x 2 + 10x + 9x + 90 x 2 + 9x + 20 = x 2 + 19x + 90 9x + 20 = 19x + 90 Combine like terms. Subtract x 2 from both sides. Use the distributive property to remove parentheses.

34. 34 Example – Solution 20 = 10x + 90 –70 = 10x –7 = x Check: (x + 5)(x + 4) = (x + 9)(x + 10) (–7 + 5)(–7 + 4) ≟ (–7 + 9)(–7 + 10) (–2)(–3) ≟ (2)(3) 6 = 6 Since the result checks, the solution is –7. Subtract 9x from both sides. Subtract 90 from both sides. Divide both sides by 10. Replace x with –7. Do the additions within parentheses. cont’d

35. 35 Solve an application involving multiplication of polynomials 6.

36. 36 Example – Dimensions of a Painting A square painting is surrounded by a border 2 inches wide. If the area of the border is 96 square inches, find the dimensions of the painting. Analyze the problem Refer to Figure 4-6, which shows a square painting surrounded by a border 2 inches wide. Figure 4-6

37. 37 Example – Dimensions of a Painting We can let x represent the length in inches of each side of the square painting. The outer rectangle is also a square, and one length is (x + 2 + 2) or (x + 4) inches. Form an equation We know that the area of the border is 96 square inches, the area of the larger square is (x + 4)(x + 4), and the area of the painting is x  x. cont’d

38. 38 Example – Dimensions of a Painting If we subtract the area of the painting from the area of the larger square, the difference is 96 (the area of the border). Solve the equation (x + 4)(x + 4) – x 2 = 96 x 2 + 8x + 16 – x 2 = 96 8x + 16 = 96 Use the distributive property to remove parentheses. Combine like terms. cont’d

39. 39 Example – Dimensions of a Painting 8x = 80 x = 10 State the conclusion The dimensions of the painting are 10 inches by 10 inches. Check the result Check the result. Subtract 16 from both sides. Divide both sides by 8. cont’d