Zeros of Polynomials Lesson 4.4.

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Presentation transcript:

Zeros of Polynomials Lesson 4.4

Factor Theorem 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) – 33

Example 2: 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

Reduced Polynomial Theorem 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 + 2 4x3 – 12x2 – 19x + 42 ÷ x + 2 4x2 – 20x + 21 so (2x – 7) (2x – 3) x = 7/2 and x = 3/2

Number of Zeros of a Polynomial 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.

Multiplicity of zero 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

Homework Pages 245 – 247 1 – 9, 13