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Suppose that the polynomial function [tex]f[/tex] is defined as follows.
[tex]\[
f(x) = 2(x + 4)^2(x - 5)^2(x - 7)^3
\][/tex]

List each zero of [tex]f[/tex] according to its multiplicity in the categories below.

If there is more than one answer for a multiplicity, separate them with a comma.

Zero(s) of multiplicity one:

Zero(s) of multiplicity two:

Zero(s) of multiplicity three:

Sagot :

GinnyAnswer
Let's analyze the polynomial function \( f(x) \) given by:
[tex]\[ f(x) = 2(x+4)^2(x-5)^2(x-7)^3 \][/tex]

To find the zeros and their multiplicities, we will examine each factor in the polynomial.

1. Identifying Zeros:
- From the factor \((x+4)^2\), set \( x+4 = 0 \):
[tex]\[ x = -4 \][/tex]
- From the factor \((x-5)^2\), set \( x-5 = 0 \):
[tex]\[ x = 5 \][/tex]
- From the factor \((x-7)^3\), set \( x-7 = 0 \):
[tex]\[ x = 7 \][/tex]

2. Determining Multiplicities:
- The factor \((x+4)^2\) has a zero at \( x = -4 \) with multiplicity 2.
- The factor \((x-5)^2\) has a zero at \( x = 5 \) with multiplicity 2.
- The factor \((x-7)^3\) has a zero at \( x = 7 \) with multiplicity 3.

3. Categorize Each Zero According to Its Multiplicity:
- Zero(s) of multiplicity one:
There are no zeros in the polynomial \( f(x) \) with a multiplicity of one.

- Zero(s) of multiplicity two:
The zeros \( x = -4 \) and \( x = 5 \) each have a multiplicity of two.

- Zero(s) of multiplicity three:
The zero \( x = 7 \) has a multiplicity of three.

In summary:

- Zero(s) of multiplicity one: \(\) (no zeros)
- Zero(s) of multiplicity two: \(-4, 5\)
- Zero(s) of multiplicity three: \(7\)

So the answer is:

- Zero(s) of multiplicity one:
- Zero(s) of multiplicity two: \(-4, 5\)
- Zero(s) of multiplicity three: [tex]\(7\)[/tex]
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