Polynomials: Multiplicity of a Zero

1. Polynomials: Multiplicity of a Zero The polynomial, P(x) = x3 – x2 – 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. 80 70 60 50 40 Negative five is a zero of odd multiplicity so the graph crosses at (- 5, 0). 30 20 10 0 -6 -5 -4 -3 -2 -1 1 2 3 4 Three is a zero of even multiplicity so the graph touches but does not cross at (3, 0). -10 -20 Polynomials: Multiplicity of a Zero 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). The graph of P(x) = x3 – x2 – 21x + 45 illustrates these properties of zeros. Slide 2

3. zero multiplicity 1 2 - 0.5 1 i 1 - i 1 There is a total of 5 zeros "counting the multiplicities". Polynomials: Multiplicity of a Zero A polynomial of degree n with real coefficients has a total of n zeros "counting the multiplicities". For example, P(x) = 2x5 – 3x4 + 2x3 – 2x2 + 1 has 5 zeros (since its degree is 5). Here is a list of its zeros. Slide 3

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