1. Polynomial & Synthetic Division Algebra III, Sec. 2.3 Objective Use long division and synthetic division to divide polynomials by other polynomials.

2. Important Vocabulary  Improper – the degree of the numerator is greater than or equal to the degree of the denominator  Proper – the degree of the numerator is less than the degree of the denominator

3. Long Division of Polynomials  Dividing polynomials is useful when ________________________.  When dividing a polynomial f(x) by another polynomial d(x), if the remainder r(x) = 0, d(x) _____________________ into f(x).  The result of a division problem can be checked by _________________. evaluating functions divides evenly multiplication

4. Example (on your handout) Divide 3x 3 + 4x – 2 by x 2 + 2x + 1. What can I multiply x 2 by to get 3x 3 ? Multiply 3x by x 2 + 2x + 1 Subtract What can I multiply x 2 by to get -6x 2 ? Multiply -6 by x 2 + 2x + 1 Subtract

5. Synthetic Divison Synthetic division is a shortcut for long division of polynomials by divisors of the form x – k.

6. Example 4 Use synthetic division to divide 2x 3 – 8x 2 + 13x – 10 by x – 2. remainder

7. Example (on your handout) Use synthetic division to divide 2x 4 + 5x 2 – 3 by x – 5. ** If powers of x are missing from the dividend, there must be a placeholder in the synthetic division for each missing term.

8. The Remainder & Factor Theorem  The Remainder Theorem states that If a polynomial f(x) is divided by x – k, the remainder is r = f(k).  To evaluate a polynomial function when x = k, divide the function by x – k (synthetic division).

9. Example 5 Use the Remainder Theorem to evaluate f(x) = 4x 3 + 10x 2 – 3x – 8 at the value f(½). Perform synthetic division.

10. Example 5 Use the Remainder Theorem to evaluate f(x) = 4x 3 + 10x 2 – 3x – 8 at the value f(½). Check with substitution. ✔

11. Example (on your handout) Use the Remainder Theorem to evaluate f(x) = 2x 4 + 5x 2 – 3 at the value x = 5. Perform synthetic division. Check with substitution. ✔

12.  The Factor Theorem states that... a polynomial f(x) has a factor (x – k) iff f(k) = 0  To use the Factor Theorem to show that (x − k) is a factor of a polynomial function f(x),... If the result is 0, then (x – k) is a factor

13. Example 6 Show that (x + 3) is a factor of f(x) = 3x 3 + 7x 2 – 3x + 9. Then find the remaining factor of f(x).

14. What have you learned? List three facts about the remainder r, obtained in the synthetic division of f(x) by x − k: