Lesson 4.4

For all polynomials p(x), x – c is a factor of p(x) if and only if p(c) = 0.

Use the Factor theorem to show that -4x + 3 is a factor of the polynomial p(x) = -4x4 + 31x3 + 3x2 + 26x – 33

Zero of -4x + 3 ¾

-4(3/4)4 + 31(3/4)3 + 3(3/4)2 + 26(3/4) – 330

Prove: For any positive integer n, show that x – 1 is a factor of x2n – 1.

This is true only if p(1) = 0

12n – 1 1 raised to any power is 1

so 12n – 1 = 0

If c1 is a zero of a polynomial p(x) and c2 is a zero of the quotient polynomial q(x) obtained when p(x) is divided by x – c1, then c2 is a zero of p(x).

Find all the zeros of the function p, where P(x) = 4x3 – 12x2 – 19x + 42

Graph it and find a zero! -2 so x + 24x3 – 12x2 – 19x + 42 ÷ x + 2

4x2 – 20x + 21 so (2x – 7) (2x – 3)

x = 7/2 and x = 3/2

A polynomial of degree n has at most n zeros.

Polynomial Graph Wiggliness Theorem:Let p(x) be a polynomial of degree ≥ 1with

real coefficients. The graph of y = p(x) can cross any horizontal line y = k at most n times.

The highest power of (x – r) that appears as a factor of that polynomial.

(x + 2)4(x – 3)( 3x + 5)

The zero x = -2 has multiplicity 4

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