1. Zeros of Polynomial Functions A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

2. The Fundamental Theorem of Algebra  The degree tells the total number of real and nonreal zeros of a polynomial function  Double solution : occurs when the graph touches the x-axis but does not cross (counts as two solutions)  Use this theorem to describe the roots of a polynomial function  The degree tells the total number of real and nonreal zeros of a polynomial function  Double solution : occurs when the graph touches the x-axis but does not cross (counts as two solutions)  Use this theorem to describe the roots of a polynomial function

3. State the number of positive real zeros, negative real zeros and imaginary zeros of each function.

4. Synthetic Division  Used to divide polynomials  Opposite of linear term goes in “box”  Coefficients go across top  Bring down first term and multiply by “box”  Write under next term and add  Repeat process  Used to divide polynomials  Opposite of linear term goes in “box”  Coefficients go across top  Bring down first term and multiply by “box”  Write under next term and add  Repeat process

5. Dividing Polynomials

6. Rational Root Theorem

7. Finding Actual Rational Roots

8. Find all zeros for polynomial functions  Find possible rational roots  Find actual rational roots  Use rational roots and synthetic division to simplify equation into a quadratic  Use quadratic formula or square roots to find complex and radical solutions  Find possible rational roots  Find actual rational roots  Use rational roots and synthetic division to simplify equation into a quadratic  Use quadratic formula or square roots to find complex and radical solutions

9. Examples