2Factor TheoremFor 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) = -4x x3 + 3x2 + 26x – 33Zero of -4x + 3 ¾-4(3/4)4 + 31(3/4)3 + 3(3/4)2 + 26(3/4) – 33
3Example 2:Prove: For any positive integer n, show that x – 1 is a factor of x2n – 1.This is true only if p(1) = 012n – 11 raised to any power is 1so 12n – 1 = 0
4Reduced 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, whereP(x) = 4x3 – 12x2 – 19x + 42Graph it and find a zero! -2 so x + 24x3 – 12x2 – 19x + 42 ÷ x + 24x2 – 20x + 21so (2x – 7) (2x – 3)x = 7/2 and x = 3/2
5Number 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.
6Multiplicity of zeroThe 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