1. 10-6 Dividing Polynomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 10-6 Dividing Polynomials Warm Up Divide. 1. m 2 n ÷ mn 4 2. 2x 3 y 2 ÷ 6xy 3. (3a + 6a 2 ) ÷ 3a 2 b Factor each expression. 4. 5x 2 + 16x + 12 5. 16p 2 – 72p + 81

3. 10-6 Dividing Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, by using these techniques. 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

4. 10-6 Dividing Polynomials To divide a polynomial by a monomial, you can first write the division as a rational expression. Then divide each term in the polynomial by the monomial.

5. 10-6 Dividing Polynomials Additional Example 1: Dividing a Polynomial by a Monomial Divide (5x 3 – 20x 2 + 30x) ÷ 5x x 2 – 4x + 6 Write as a rational expression. Divide each term in the polynomial by the monomial 5x. Divide out common factors. Simplify.

6. 10-6 Dividing Polynomials Check It Out! Example 1a Divide. (8p 3 – 4p 2 + 12p) ÷ ( – 4p 2 ) Write as a rational expression. Divide each term in the polynomial by the monomial –4p 2. Divide out common factors. Simplify.

7. 10-6 Dividing Polynomials Check It Out! Example 1b Divide. (6x 3 + 2x – 15) ÷ 6x Write as a rational expression. Divide each term in the polynomial by the monomial 6x. Divide out common factors in each term. Simplify.

8. 10-6 Dividing Polynomials Division of a polynomial by a binomial is similar to division of whole numbers.

9. 10-6 Dividing Polynomials Additional Example 2A: Dividing a Polynomial by a Binomial Divide. x + 5 Factor the numerator. Divide out common factors. Simplify.

10. 10-6 Dividing Polynomials Additional Example 2B: Dividing a Polynomial by a Binomial Divide. Factor both the numerator and denominator. Divide out common factors. Simplify.

11. 10-6 Dividing Polynomials Put each term of the numerator over the denominator only when the denominator is a monomial. If the denominator is a polynomial, try to factor first. Helpful Hint

12. 10-6 Dividing Polynomials Check It Out! Example 2a Divide. k + 5 Factor the numerator. Divide out common factors. Simplify.

13. 10-6 Dividing Polynomials Check It Out! Example 2b Divide. b – 7 Factor the numerator. Divide out common factors. Simplify.

14. 10-6 Dividing Polynomials Check It Out! Example 2c Divide. s + 6 Factor the numerator. Divide out common factors. Simplify.

15. 10-6 Dividing Polynomials Recall how you used long division to divide whole numbers as shown at right. You can also use long division to divide polynomials. An example is shown below. ) x 2 + 3x + 2 x + 1 x 2 + 2x x + 2 0 (x 2 + 3x + 2) ÷ (x + 2) Divisor Quotient Dividend

16. 10-6 Dividing Polynomials Using Long Division to Divide a Polynomial by a Binomial Step 1 Write the binomial and polynomial in standard form. Step 3 Multiply this first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms. Step 2 Divide the first term of the dividend by the first term of the divisor. This the first term of the quotient. Step 4 Subtract the product from the dividend. Step 5 Bring down the next term in the dividend. Step 6 Repeat Steps 2-5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial.

17. 10-6 Dividing Polynomials Additional Example 3A: Polynomial Long Division Divide using long division. Check your answer. (x 2 +10x + 21) ÷ (x + 3) x 2 + 10x + 21 ) Step 1x + 3 Write in long division form with expressions in standard form. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. x 2 + 10x + 21 ) Step 2x + 3 x

18. 10-6 Dividing Polynomials Additional Example 3A Continued Divide using long division. (x 2 +10x + 21) ÷ (x + 3) Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. x 2 + 10x + 21 ) Step 3x + 3 x x 2 + 3x x 2 + 10x + 21 ) Step 4x + 3 – (x 2 + 3x) x 0 + 7x Subtract the product from the dividend.

19. 10-6 Dividing Polynomials Additional Example 3A Continued Divide using long division. x 2 + 10x + 21 ) Step 5x + 3 – (x 2 + 3x) x + 21 Bring down the next term in the dividend. Repeat Steps 2-5 as necessary. x 2 + 10x + 21 ) Step 6x + 3 – (x 2 + 3x) x + 7 7x + 21 – (7x + 21) 0 The remainder is 0. 7x7x

20. 10-6 Dividing Polynomials Additional Example 3A Continued Check: Multiply the answer and the divisor. (x + 3)(x + 7) x 2 + 7x + 3x + 21 x 2 + 10x + 21

21. 10-6 Dividing Polynomials When the remainder is 0, you can check your simplified answer by multiplying it by the divisor. You should get the numerator. Helpful Hint

22. 10-6 Dividing Polynomials Additional Example 3B: Polynomial Long Division Divide using long division. x 2 – 2x – 8 ) x – 4 Write in long division form. – (x 2 – 4x) 2x2x x 2 – 2x – 8 ) x – 4 – (2x – 8) 0 x 2 ÷ x = x Multiply x  (x – 4). Subtract. Bring down the 8. 2x ÷ x = 2. Multiply 2(x – 4). Subtract. The remainder is 0. x + 2 – 8

23. 10-6 Dividing Polynomials Additional Example 3B Continued Check: Multiply the answer and the divisor. (x + 2)(x – 4) x 2 – 4x + 2x – 8 x 2 – 2x + 8

24. 10-6 Dividing Polynomials Check It Out! Example 3a Divide using long division. (2y 2 – 5y – 3) ÷ (y – 3) 2y 2 – 5y – 3 ) Step 1 y – 3 Write in long division form with expressions in standard form. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. 2y 2 – 5y – 3 ) Step 2 y – 3 2y2y

25. 10-6 Dividing Polynomials Check It Out! Example 3a Continued Divide using long division. (2y 2 – 5y – 3) ÷ (y – 3) Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. Subtract the product from the dividend. 2y 2 – 5y – 3 ) Step 3 y – 3 2y2y 2y 2 – 6y – (2y 2 – 6y) 0 + y 2y 2 – 5y – 3 ) Step 4 y – 3 2y2y

26. 10-6 Dividing Polynomials Check It Out! Example 3a Continued Divide using long division. ) Step 5 y – 3 2y2y – 3– 3 Bring down the next term in the dividend. Repeat Steps 2–5 as necessary. 2y2y 2y 2 – 5y – 3 ) Step 6 y – 3 – (2y 2 – 6y) y – 3 – (y – 3) 0 The remainder is 0. 2y 2 – 5y – 3 – (2y 2 – 6y) y + 1

27. 10-6 Dividing Polynomials Check: Multiply the answer and the divisor. (y – 3)(2y + 1) 2y 2 + y – 6y – 3 2y 2 – 5y – 3 Check It Out! Example 3a Continued

28. 10-6 Dividing Polynomials Check It Out! Example 3b Divide using long division. (a 2 – 8a + 12) ÷ (a – 6) a 2 – 8a + 12 ) a – 6 Write in long division form. – (a 2 – 6a) –2a–2a a a 2 – 8a + 12 ) a – 6 – ( – 2a + 12) 0 a 2 ÷ a = a Multiply a  (a – 6). Subtract. Bring down the 12. –2a ÷ a = –2. Multiply –2(a – 6). Subtract. The remainder is 0. – 2 + 12

29. 10-6 Dividing Polynomials Check It Out! Example 3b Continued Check: Multiply the answer and the divisor. (a – 6)(a – 2) a 2 – 2a – 6a + 12 a 2 – 8a + 12

30. 10-6 Dividing Polynomials Sometimes the divisor is not a factor of the dividend, so the remainder is not 0. Then the remainder can be written as a rational expression.

31. 10-6 Dividing Polynomials Additional Example 4: Long Division with a Remainder Divide (3x 2 + 19x + 26) ÷ (x + 5) 3x 2 + 19x + 26 ) x + 5 Write in long division form. 3x 2 + 19x + 26 ) x + 5 3x3x – (3x 2 + 15x) 4x4x 3x 2 ÷ x = 3x. Multiply 3x(x + 5). Subtract. Bring down the 26. 4x ÷ x = 4. Multiply 4(x + 5). Subtract. – (4x + 20) 6 The remainder is 6. Write the remainder as a rational expression using the divisor as the denominator. + 4 + 26

32. 10-6 Dividing Polynomials Additional Example 4 Continued Divide (3x 2 + 19x + 26) ÷ (x + 5) Write the quotient with the remainder.

33. 10-6 Dividing Polynomials Check It Out! Example 4a Divide. 3m 2 + 4m – 2 ) m + 3 Write in long division form. 3m 2 + 4m – 2 ) m + 3 3m3m – (3m 2 + 9m) 3m 2 ÷ m = 3m. Multiply 3m(m + 3). Subtract. Bring down the –2. –5m ÷ m = –5. Multiply –5(m + 3). Subtract. –5m–5m The remainder is 13. 13 – ( – 5m – 15) – 5 – 2

34. 10-6 Dividing Polynomials Check It Out! Example 4a Continued Divide. Write the remainder as a rational expression using the divisor as the denominator.

35. 10-6 Dividing Polynomials Check It Out! Example 4b Divide. y 2 + 3y + 2 ) y – 3 Write in long division form. – (y 2 – 3y) y 2 ÷ y = y. Multiply y(y – 3). Subtract. Bring down the 2. 6y ÷ y = 6. y y 2 + 3y + 2 ) y – 3 Multiply 6(y – 3). Subtract. The remainder is 20. 20 6y6y – (6y – 18) Write the quotient with the remainder. + 6 + 2 y + 6 +

36. 10-6 Dividing Polynomials Sometimes you need to write a placeholder for a term using a zero coefficient. This is best seen if you write the polynomials in standard form.

37. 10-6 Dividing Polynomials Additional Example 5: Dividing Polynomials That Have a Zero Coefficient Divide (x 3 – 7 – 4x) ÷ (x – 3). x 3 + 0x 2 – 4x – 7 ) x – 3 x 3 ÷ x = x 2 Multiply x 2 (x – 3). Subtract. (x 3 – 4x – 7) ÷ (x – 3) Write the polynomials in standard form. Write in long division form. Use 0x 2 as a placeholder for the x 2 term. x2x2 x 3 + 0x 2 – 4x – 7 ) x – 3 –(x3 – 3x2)–(x3 – 3x2) 3x23x2 – 4x Bring down –4x.

38. 10-6 Dividing Polynomials Additional Example 5 Continued x 3 + 0x 2 – 4x – 7 ) x – 3 3x 3 ÷ x = 3x Multiply x 2 (x – 3). Subtract. x2x2 –(x3 – 3x2)–(x3 – 3x2) 3x23x2 – 4xBring down – 4x. – (3x 2 – 9x) 5x5x – (5x – 15) 8 Bring down – 7. Multiply 3x(x – 3). Subtract. The remainder is 8. + 3x – 7 Multiply 5(x – 3). Subtract. + 5 (x 3 – 4x – 7) ÷ (x – 3) =

39. 10-6 Dividing Polynomials Recall from Chapter 7 that a polynomial in one variable is written in standard form when the degrees of the terms go from greatest to least. Remember!

40. 10-6 Dividing Polynomials Divide (1 – 4x 2 + x 3 ) ÷ (x – 2). Check It Out! Example 5a (x 3 – 4x 2 + 1) ÷ (x – 2) x 3 – 4x 2 + 0x + 1x – 2 ) Write in standard form. Write in long division form. Use 0x as a placeholder for the x term. x 3 – 4x 2 + 0x + 1x – 2 ) x2x2 x 3 ÷ x = x 2 – ( – 2x 2 + 4x) – 4x – ( – 4x + 8) –7–7 Bring down 0x. – 2x 2 ÷ x = –2x. Multiply –2x(x – 2). Subtract. Bring down 1. Multiply –4(x – 2). Subtract. – (x 3 – 2x 2 ) – 2x 2 Multiply x 2 (x – 2). Subtract. – 2x + 0x + 1 – 4

41. 10-6 Dividing Polynomials Divide (1 – 4x 2 + x 3 ) ÷ (x – 2). Check It Out! Example 5a Continued (1 – 4x 2 + x 3 ) ÷ (x – 2) =

42. 10-6 Dividing Polynomials Divide (4p – 1 + 2p 3 ) ÷ (p + 1). Check It Out! Example 5b (2p 3 + 4p – 1) ÷ (p + 1) 2p 3 + 0p 2 + 4p – 1 p + 1 ) Write in standard form. Write in long division form. Use 0p 2 as a placeholder for the p 2 term. 2p 3 + 0p 2 + 4p – 1 p + 1 ) 2p22p2 p 3 ÷ p = p 2 – ( – 2p 2 – 2p) 6p – (6p + 6) –7–7 Bring down 4p. –2p 2 ÷ p = –2p. Multiply –2p(p + 1). Subtract. Bring down –1. Multiply 6(p + 1). Subtract. – (2p 3 + 2p 2 ) – 2p 2 Multiply 2p 2 (p + 1). Subtract. – 2p + 4p – 1 + 6

43. 10-6 Dividing Polynomials Check It Out! Example 5b Continued (2p 3 + 4p – 1) ÷ (p + 1) =

44. 10-6 Dividing Polynomials Lesson Quiz: Part I Add or Subtract. Simplify your answer. 1. 3. 2. (12x 2 – 4x 2 + 20x) ÷ 4x 3x 2 – x + 5 x – 2 4. x + 3 2x + 3

45. 10-6 Dividing Polynomials Lesson Quiz: Part II Divide using long division. 5. 6. (8x 2 + 2x 3 + 7)  (x + 3) (x 2 + 4x + 7)  (x + 1)