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Polynomial Division: Long & Synthetic Methods

Learn to divide polynomials using long division and synthetic methods. Understand remainder and factor theorems. Practice with examples and check points. Dive into the Synthetic Division process step by step. Apply Remainder Theorem and Factor Theorem for polynomial division.

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Polynomial Division: Long & Synthetic Methods

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  1. Lesson 2.4, page 301Dividing Polynomials Objective: To divide polynomials using long and synthetic division, and to use the remainder and factor theorems.

  2. How do you divide a polynomial by another polynomial? • Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term. • Work from left to right, starting with the highest degree term. • Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly.

  3. Dividing a Poly by a Binomial

  4. See Example 1, page 302. • Check Point 1:(x2 + 14x + 45)  (x + 9)

  5. Check Point 2(7 – 11x - 3x2 + 2x3)  (x - 3)

  6. Missing Terms? • Write the polynomial in standard form. • If any power is missing, use a zero to hold the place of that term. • Divide as before.

  7. Check Point 3(2x4 + 3x3 – 7x - 10)  (x2 – 2x)

  8. Synthetic Division • a simpler process (than long division) for dividing a polynomial by a binomial; uses coefficients and part of the divisor See Example 4, page 306.

  9. STEPS for Synthetic Division, pg. 306 1) Write polynomial in descending order of the degrees. 2) List the coefficients. (If one power is missing, put a zero to hold that place.) 3) Write the constant cof the divisor x - c to the left. 4) Bring down the first coefficient. 5) Multiply the first coefficient by c, write the product under the 2nd coefficient and add. 6) Multiply this sum by c, write it under the next coefficient and add. Repeat until all coefficients have been used. 7) The numbers on the bottom row are the coefficients of the answer. The first power on the variable will be one less than the highest power in the original polynomial.

  10. Check Point 4, page 307 • Use Synthetic Division: x3 – 7x – 6 by x + 2. Caution: What is missing?

  11. The Remainder Theorem, pg. 307 If the polynomial f(x) is divided by x – c, then the remainder is the same value as f(c). Also: f(x) = (x – c) q(x) + r divisor (quotient) + remainder

  12. See Example 5, pg. 308 • Check Point 5: Given f(x) = 3x3 + 4x2 – 5x + 3,use the remainder theorem to find f(- 4).

  13. Dividing a Poly by a Binomial • If a binomial divides into a polynomial with no remainder, the binomial is a factor of the polynomial.

  14. Factor Theorem, pg. 308 For the polynomial f(x), if f(c) = 0, then x – c is a factor of f(x) Remember . . . If something is a factor, then it divides the term evenly with 0 remainder.

  15. See Example 6, pg. 309. • Check Point 6: Solve the equation 15x3 + 14x2 – 3x – 2 = 0, given that -1 is a zero of f(x) = 15x3 + 14x2 – 3x – 2.

  16. Determine if -1 is a zero ofg(x) = x4 - 6x3 + x2 + 24x -20.

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