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

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

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

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

5. Example 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.

6. Checkpoint Use synthetic division to divide 5x 3 + 8x 2 – x + 6 by x + 2. 5x 2 – 2x + 3

7. 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).

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

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

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

11. Checkpoint Use the Remainder Theorem to find each function value given. f(x) = 4x 3 + 10x 2 – 3x – 8 a.) f(-1) b.) f(4) c.) f(1/2) d.) f(-3) (a)1 (b) 396 (c)-13/2 (d) -17

12. Factor Theorem  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 Show that (x + 3) is a factor of f(x) = 3x 3 + 7x 2 – 3x + 9. Then find the remaining factor of f(x). prime

14. Checkpoint Show that (x + 3) is a factor of f(x)= x 3 – 19x – 30. Then find the remaining factors of f(x). f(-3)=0 f(x)= (x + 3)(x – 5)(x + 2)