 Section 3.3 Real Zeros of Polynomial Functions. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational.

Presentation on theme: "Section 3.3 Real Zeros of Polynomial Functions. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational."— Presentation transcript:

Section 3.3 Real Zeros of Polynomial Functions

Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational Zeros Theorem – Find the Real Zeros of a Polynomial Function – Solve Polynomial Equations – Use the Theorem for Bounds on Zeros – Use the Intermediate Value Theorem

Synthetic and Long Division of Polynomials EXAMPLES

Remainder Theorem Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c). Find the remainder if is divided by What is f(-3)?

Factor Theorem Let f be a polynomial function. Then x-c is a factor of f(x) if and only if f(c)=0. So this means that the remainder when the polynomial is divided by x-c is 0. Thus x-c divides into the polynomial evenly. Theorem: A polynomial function of degree n has at most n real zeros.

Rational Zeros Theorem Let f be a polynomial function of degree 1 or higher of the form Where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of and q must be a factor of. We can test each possible solution with synthetic division.

Bounds on Zeros f(x) is a polynomial function whose leading coefficient is 1 A bound is the smaller of the following two numbers: OR … Write the bound as plus or minus.

Finding Zeros of Polynomials Use the Rational Zeros Theorem and repeated division to find the zeros EXAMPLES

Download ppt "Section 3.3 Real Zeros of Polynomial Functions. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational."

Similar presentations