1. Division Algorithm

1. 1. Division Algorithm Rewrite as dividend remainder = quotient + divisor divisor + remainder = x dividend divisor quotient

2. Review… synthetic division Use synthetic division to evaluate each: 1) a) What is the remainder ? b) Is (x-1) a factor of f(x) ? c) What is the value of f(1)?

3. 3.6 The Real Zeros of Polynomial Functions • Remainder and Factor Theorem • Rational Zeros Theorem • Find Real zeros of a polynomial • Solve polynomial equation

4. 1. Division Algorithm Compute dividend remainder = quotient + divisor divisor

5. 1. Division Algorithm x + = quotient remainder dividend divisor Notice: f(1) and the remainder are the same

6. 1. Remainder Theorem If a polynomial f(x) is divided by (x-c) then the remainder is the number f (c). Determine the remainder of two ways 1) Synthetic Division OR 2) Evaluate f(1)

7. What is a factor? What is the remainder of

8. 2. Factoring of Polynomials Goal: determine if is a factor of Two ways to determine if x-c is a factor of f: 1) Use synthetic division and check if remainder is zero. OR 2) Determine if f(c)=0 Is x + 1 is a factor ? x – 1 ?

9. 2. Factoring of Polynomials To write f(x) in factored form. If f(c)=0, then (x-c) is a factor Write f(x) = q(x)(x-c)

10. 4. Rational Zeros Theorem Recall: Rational numbers can be written as a fraction Goal: Make a list of all possible rational zeros for a polynomial. is a polynomial with integer coefficients. possible rational zeros = where p = factors of constant term q = factors of leading coefficient List all possible rational zeros for

11. Potential Rational Zeros Number of Real Zeros Theorem A polynomial of degree n, has at most n real zeros.

12. Factor a polynomial • List potential rational zeros( ) • If using a graphing calculator: • graph it and test potential zeros using TRACE or Table • If do not have graphing calculator, use trial and error: • synthetic division for each potential zero, until a zero is found • OR check each potential zero for f (c) = 0. • 5. Each time a zero is found: • Find q(x): Synthetic division with (x-c). • Now write: f(x) = (x – c) q(x). • 6. Set q(x) = 0 and solve if it is quadratic. Otherwise, repeat steps 2-5 and append result to #5..

13. Real Zeros of Polynomials Find the real zeros of f and factor f over the reals.

14. Real Zeros of Polynomials Find the real zeros of f and factor f over the reals.

15. Practice Problems Find the real zeros of f and factor f over the reals.