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section4_2

# section4_2 - Section 4.2 Polynomial Long Division and...

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Unformatted text preview: Section 4.2 Polynomial Long Division and Synthetic Division Page 1 of 17 Example 1: 2 x4 − 8 x2 + 3x − 5 Find the quotient and remainder for by using polynomial x−5 long division. Solution to Example 1: The dividend is the polynomial P ( x ) = 2 x 4 − 8 x 2 + 3x − 5 and the divisior is the polynomial D( x) = x − 5. Section 4.2 Polynomial Long Division and Synthetic Division Page 2 of 17 Example 2: Synthetic division is a quick method of dividing polynomials, which can be used when the divisor is of the form D( x) = x − c. Find the quotient and remainder for 2 x 4 − 8 x 2 + 3x − 5 by using synthetic division. x −5 Solution to Example 2: The dividend is the polynomial P ( x ) = 2 x 4 − 8 x 2 + 3x − 5 and the divisior is the polynomial D( x) = x − 5. The divisor is of the form D( x) = x − c, where c = 5. Section 4.2 Polynomial Long Division and Synthetic Division Page 3 of 17 Example 3: Find the quotient and remainder for 3 3 5 2 21 by using synthetic division. 5o9cn0-1.6TDTc05Tj/619tx94-02.53()Tn1 Section 4.2 Polynomial Long Division and Synthetic Division Page 4 of 17 Example 5: Solution to Example 5: Example 6: Section 4.2 Polynomial Long Division and Synthetic Division Page 5 of 17 Solution to Example 6: Section 4.2 Polynomial Long Division and Synthetic Division Page 6 of 17 Example 7: Solution to Example 7: Section 4.2 Polynomial Long Division and Synthetic Division Page 7 of 17 Section 4.2 The Remainder and Factor Theorems Page 8 of 17 Example 1: Use synthetic division and the Remainder Theorem to evaluate P (−4) for P ( x) = 2 x3 − 2 x 2 + 11x − 100. Solution to Example 1: By the Remainder Theorem, P( −4) is the remainder when P( x) = 2 x3 − 2 x 2 + 11x − 100 is divided by x − (−4) = x + 4. Example 2: Use the Factor Theorem to show that x − 2 is a factor of the polynomial P ( x) = x5 − 13 x 4 + 57 x 3 − 83 x 2 − 34 x + 120. Solution to Example 2: Section 4.2 The Remainder and Factor Theorems Page 9 of 17 Example 3: Find a polynomial of degree three that has zeros − 2, −1, and 3. Solution to Example 3: Section 4.2 The Remainder and Factor Theorems Page 10 of 17 Example 5: Solution to Example 5: Section 4.2 The Remainder and Factor Theorems Page 11 of 17 Example 6: Solution to Example 6: Section 4.2 The Remainder and Factor Theorems Page 12 of 17 Example 7: Solution to Example 7: Section 4.2 The Remainder and Factor Theorems Page 13 of 17 Section 4.2 The Remainder and Factor Theorems Page 14 of 17 Example 8: Solution to Example 8: Section 4.2 The Remainder and Factor Theorems Page 15 of 17 Exercise Set 4.2: Dividing Polynomials Use long division to find the quotient and the remainder. 1. x3 − 2 x 2 − 19 x − 9 x+3 15. x 4 + 3x 2 − 4 x +1 2 x5 + 3x 4 − 7 x + 8 x −1 3 x 4 − 11x3 − 27 x 2 + 18 x + 10 x −5 2 x 4 + 3 x3 − 18 x 2 + 5 x − 12 x−2 x3 + 8 x+2 x 4 − 81 x+3 4 x3 − 7 x + 5 x− 1 2 6 x 4 + x3 − 10 x 2 + 9 x− 1 3 16. 2. x3 − 2 x 2 − 22 x + 33 x−5 6 x3 + 5 x 2 + 6 x − 12 2x −1 12 x3 + 13 x 2 − 22 x − 14 3x + 4 2 x3 + 13x 2 + 28 x + 21 x 2 + 3x + 1 17. 3. 18. 4. 19. 5. 20. 6. x 4 − 7 x3 + 4 x 2 − 42 x + 12 x2 − 7 x − 2 21. 7. 2 x5 + 32 x 4 + 3 x3 + 44 x 2 − 14 4x2 + 6 22. 8. 10 x8 + 20 x 6 + x 4 + 2 x3 + 28 x 2 + 4 2x4 + 6x2 + 3 9. 3 x 4 − x3 − 15 x2 + 5 3 Evaluate P(c) by using synthetic division and the Remainder Theorem. 23. P ( x) = x3 − 4 x 2 + 5 x + 2; c = 2 24. 24. () x P5 3 10. 3x − 4 x − 2 x + 7 x2 − 2 x 5 + 7 x52x− 8 − 3; 2 = −1 Use synthetic division to find the quotient and the remainder. 11. x2 − 8x + 4 x − 10 x2 − 4 x − 6 x+3 3 x3 + 13 x 2 − 6 x + 28 x+5 + 12. 13. 2 x3 − x 2 − 31x 14. x−4 4 Exercise Set 4.2: Dividing Polynomials Answer the following. 31. Use the Factor Theorem to show that x − 3 is a factor of 5 29 15 ...
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