If f(x) is a polynomial of degree n, where n>0, then f has ...

Section 2.5: Zeros of Polynomial Functions

If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system.

Example 1-Zeros of Polynomial Functions

If f(x) is a polynomial of degree n, where n>0, then f has precisely n linear factors Linear Factorization Theorem

The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system

The Rational Zero Test

Example 2-Rational Zeros with Leading Coefficient of 1

Example 3-Rational Zeros with Leading Coefficient of 1

Example 4-Using the Rational Zero Test

Example 5-Solving a Polynomial Equation

Conjugate PairsLet f(x) be a polynomial function that has real coefficients. If a+bi, where b=0, is a zero of the function, the conjugate a-bi is also a zero of the function.

Example 6

Factoring a PolynomialEvery polynomial of degree n>0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

A quadratic factor with no real zeros is said to be prime or irreducible over the reals.

Example 7

Example 8

Other Tests for Zeros of Polynomials

Variation in sign means that two consecutive coefficients have opposite signs.

Example 9

Example 10

Example 11

Assignment: #1, 2, 5, 6, 9, 10, 12, 19, 29, 38, 39, 43, 46, 50, 55, 62, 75, 78, 79, 85, 88, 91, 94, 108

If f(x) is a polynomial of degree n, where n>0, then f has ...

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