Question

Suppose that the polynomial function f is defined as follows.f(x)=3x³(x-4)²(x - 5)(x+7)³List each zero off according to its multiplicity in the categories below.If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, click on "None."Zero(s) of multiplicity one:Zero(s) of multiplicity two:Zero(s) of multiplicity three:08 0.0....XNoneŚ

Answer 1

f(x)=3x³(x-4)²(x - 5)(x+7)³

To find the zeros, set the function equal to zero

3x³(x-4)²(x - 5)(x+7)³ =0

Using the zero product property

3x^3 =0 ( x-4) ^2 =0 x-5 =0 ( x+7) ^3 =0

Solving each equation

x=0 x=4 x=5 x = -7

We can find the multiplicity from the power of the factor

x=0 has a multiplicity of 3

x =4 has a multiplicity of 2

x =5 has a multiplicity of 1

x = -7 has a multiplicity of 3

Zero(s) of multiplicity one:5

Zero(s) of multiplicity two:4

Zero(s) of multiplicity three: 0,-7