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Section 3.4 Zeros of Polynomial Functions *Rational Zero Theorem p p (where is reduced q q to lowest terms) is rational zero of f , then p is a factor of the constant term, a 0 , q is a factor of the leading coefficient, a n . Factors of a0 Possible rational zeros= . Factors of an n n 1 If f ( x) a n x a n 1 x a1 x a 0 has integer coefficient and 3 2 Example 1) List all possible rational zeros of f ( x) x 2 x 5x 6 . 5 4 Example 2) List all possible rational zeros of f ( x) 4 x 12 x x 3 . *Finding Zeros of a Polynomial Function Step 1 List all possible rational zeros Step 2 Use synthetic division to find a rational zero among the possibilities. Step 3 Factor the polynomial using synthetic division. Step 4 Repeat Step 1 through Step 3 until you get a quadratic factor. 3 2 Example 3) Find all zeros of f ( x) x 8 x 11x 20 . 1 3 2 Example 4) Find all zeros of f ( x) x x 5x 2 . 4 3 2 Example 5) Solve x 6 x 22 x 30 x 13 0 . 2 *Properties of Polynomial Equations 1. If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. 2. If a bi is a root of a polynomial equation with real coefficient (b 0) , then the complex imaginary number a bi is also a root. Complex imaginary roots, if they exist, occur in conjugate pairs. *The Linear Factorization Theorem n n 1 If f ( x) a n x a n 1 x a1 x a 0 , where n 1 and an 0 , then f ( x) an ( x c1 )( x c2 )( x cn ) , Where c1 , c2 ,, cn are complex numbers (possibly real and not necessarily distinct). Example 6) Find a third-degree polynomial function f (x) with real coefficients that has -3 and i as zeros and such that f (1) 8 . 3