3. Division and Factors When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor of the dividend. Example: Divide to determine whether x + 3 and x  1 are factors of

4. Division and Factors continued Divide: Since the remainder is –64, we know that x + 3 is not a factor.

5. Division and Factors continued Divide: Since the remainder is 0, we know that x  1 is a factor.

6. 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.

7. The Remainder Theorem If a number c is substituted for x in a polynomial f(x), then the result f(c) is the remainder that would be obtained by dividing f(x) by x  c. That is, if f(x) = (x  c) Q(x) + R, then f(c) = R. Synthetic division is a “collapsed” version of long division; only the coefficients of the terms are written.

8. Synthetic division is a quick form of long division for polynomials where the divisor has form x - c. In synthetic division the variables are not written, only the essential part of the long division.

9. 1 -2 3 -6 0 0 quotient remainder

10. Example Use synthetic division to find the quotient and remainder. The quotient is – 4x 4 – 7x 3 – 8x 2 – 14x – 28 and the remainder is –6. –6–28–14–8–7–4 –56–28–16–14–8 500261–42 Note: We must write a 0 for the missing term.

11. Example continued: written in the form

12. Example Determine whether 4 is a zero of f(x), where f(x) = x 3  6x 2 + 11x  6. We use synthetic division and the remainder theorem to find f(4). Since f(4)  0, the number is not a zero of f(x). 63–21 12–84 –611–614

13. The Factor Theorem For a polynomial f(x), if f(c) = 0, then x  c is a factor of f(x). Example: Let f(x) = x 3  7x + 6. Solve the equation f(x) = 0 given that x = 1 is a zero. Solution: Since x = 1 is a zero, divide synthetically by 1. Since f(1) = 0, we know that x  1 is one factor and the quotient x 2 + x  6 is another. So, f(x) = (x  1)(x + 3)(x  2). For f(x) = 0, x =  3, 1, 2. 0-611 11 6-701 1

14. Factor Theorem f(x) is a polynomial, therefore f(c) = 0 if and only if x – c is a factor of f(x). If we know a factor, we know a zero! If we know a zero, we know a factor!

15. Definition of Depressed Polynomial A Depressed Polynomial is the quotient that we get when a polynomial is divided by one of its binomial factors

16. Which of the following can be divided by the binomial factor (x - 1) to give a depressed polynomial (x - 1)? Choices: A. x2 - 2x + 1 B. x2 - 2x - 2 C. x2 - 3x - 3 D. x2 - 2

17. Using the remainder theorem to find missing coeffecients… Find the value of k that results in a remainder of “0” given…