Suppose that the polynomial function [tex]$f$[/tex] is defined as follows:
[tex]$f(x)=6(x-13)(x+12)(x-7)^2(x+8)^3$[/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 commas. If there is no answer, click on "None."
Zero(s) of multiplicity one:
Zero(s) of multiplicity two:
Zero(s) of multiplicity three:
Answer (2)
Identify the zeros of the polynomial by setting each factor to zero: x = 13 , x = − 12 , x = 7 , x = − 8 .
Determine the multiplicity of each zero by looking at the exponent of the corresponding factor.
List the zeros with multiplicity one: 13 , − 12 .
List the zeros with multiplicity two: 7 .
List the zeros with multiplicity three: − 8 .
The zeros of multiplicity one are 13 , − 12 , the zero of multiplicity two is 7 , and the zero of multiplicity three is − 8 .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = 6 ( x − 13 ) ( x + 12 ) ( x − 7 ) 2 ( x + 8 ) 3 . We need to identify the zeros of this function and their corresponding multiplicities. The zeros are the values of x that make f ( x ) = 0 . The multiplicity of a zero is the exponent of the factor that produces that zero.
Finding the Zeros To find the zeros, we set each factor equal to zero:
x − 13 = 0 ⟹ x = 13 x + 12 = 0 ⟹ x = − 12 x − 7 = 0 ⟹ x = 7 x + 8 = 0 ⟹ x = − 8
So the zeros are 13 , − 12 , 7 , and − 8 .
Determining Multiplicities Now we determine the multiplicity of each zero by looking at the exponent of its corresponding factor:
For x = 13 , the factor is ( x − 13 ) , which has an exponent of 1 . So, the multiplicity of 13 is 1 .
For x = − 12 , the factor is ( x + 12 ) , which has an exponent of 1 . So, the multiplicity of − 12 is 1 .
For x = 7 , the factor is ( x − 7 ) 2 , which has an exponent of 2 . So, the multiplicity of 7 is 2 .
For x = − 8 , the factor is ( x + 8 ) 3 , which has an exponent of 3 . So, the multiplicity of − 8 is 3 .
Listing the Zeros by Multiplicity Finally, we list the zeros according to their multiplicities:
Zero(s) of multiplicity one: 13 , − 12 Zero(s) of multiplicity two: 7 Zero(s) of multiplicity three: − 8
Examples
Understanding polynomial functions and their zeros is crucial in many areas of mathematics and engineering. For instance, when designing a bridge, engineers use polynomials to model the load distribution and identify critical points (zeros) where the structure experiences maximum stress. The multiplicity of these zeros helps determine the stability and safety margins of the bridge, ensuring it can withstand various loads and environmental conditions. By analyzing the zeros and their multiplicities, engineers can optimize the design to prevent structural failure and ensure long-term reliability.
The zeros of the polynomial function are 13 and -12 with multiplicity one, 7 with multiplicity two, and -8 with multiplicity three.
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