1. Table of Contents Polynomials: Multiplicity of a Zero The polynomial, P(x) = x 3 – x 2 – 21x + 45, factors as, P(x) = (x – 3)(x – 3)(x + 5). Its zeros (solutions of P(x) = 0) are 3 and - 5. Negative five is said to be a zero of multiplicity 1, since there is one factor, (x + 5), associated with it. Three is said to be a zero of multiplicity 2, since there are two factors, (x – 3), associated with it.

2. Table of Contents Polynomials: Multiplicity of a Zero Slide 2 The graph of P(x) = x 3 – x 2 – 21x + 45 illustrates these properties of zeros. -20 -10 0 10 20 30 40 50 60 70 80 -6-5-4-3-21234 If c is a real zero of odd multiplicity, the graph of the polynomial will cross the x-axis at (c, 0). If c is a real zero of even multiplicity, the graph of the polynomial will touch but not cross the x-axis at (c, 0). Negative five is a zero of odd multiplicity so the graph crosses at (- 5, 0). Three is a zero of even multiplicity so the graph touches but does not cross at (3, 0).

3. Table of Contents Polynomials: Multiplicity of a Zero A polynomial of degree n with real coefficients has a total of n zeros "counting the multiplicities". Slide 3 For example, P(x) = 2x 5 – 3x 4 + 2x 3 – 2x 2 + 1 has 5 zeros (since its degree is 5). Here is a list of its zeros. zero multiplicity 12 - 0.51 i1 - i1 There is a total of 5 zeros "counting the multiplicities".

4. Table of Contents Polynomials: Multiplicity of a Zero