3.3B   Zeros and Multiplicities

Zeros and Multiplicities

Identifying Zeros and Their Multiplicities

We can determine whether or not a graph will cross the x-axis by its multiplicity. Let's take a look at f(x) = 5 (x - 2) (x + 3)2 (x - 1)3 (x - 5)4

to determine its zeros and multiplicity.

(x - 2) = 0       x = 2 is a zero of multiplicity 1 because the exponent on the factor (x - 2) is 1 (odd)

(x + 3)2 = 0     x = -3 is a zero of multiplicity 2 because the exponent on the factor (x + 3) is 2 (even)

(x - 1)3 = 0      x = 1 is a zero of multiplicity 3 because the exponent on the factor (x - 1) is 3 (odd)

(x - 5)4 = 0      x = 4 is a zero of multiplicity 4 because the exponent on the factor (x - 4) is 4 (even)

   **** Adding the multiplicities (exponents) ( 1 + 2 + 3 + 4 = 10), you will obtain the degree of the polynomial.

           The maximum number of turns will be one less than the degree of the polynomial.

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Important  

*** If a zero of a function is of odd multiplicity, then the graph will cross the x-axis at that zero.

*** If a zero is of even multiplicity, then the graph will not cross the x-axis, it will just touch and turn the zero and turn away from the x-axis.