Asked by Breanna
Find all of the zeros of the polynomial function and state the multiplicity of each.
f (x) = (x^2 – 16)^2
A. – 4 with multiplicity 2 and 4 with multiplicity 2
B. – 4i with multiplicity 2 and 4i with multiplicity 2
C. 4 with multiplicity 2
D. 4 with multiplicity 4
f (x) = (x^2 – 16)^2
A. – 4 with multiplicity 2 and 4 with multiplicity 2
B. – 4i with multiplicity 2 and 4i with multiplicity 2
C. 4 with multiplicity 2
D. 4 with multiplicity 4
Answers
MathMate
f(x) = (x<sup>2</sup>-16)<sup>2</sup>
= ((x+4)(x-4))<sup>2</sup>
= (x+4)<sup>2</sup>(x-4)<sup>2</sup>
Can you take it from here?
= ((x+4)(x-4))<sup>2</sup>
= (x+4)<sup>2</sup>(x-4)<sup>2</sup>
Can you take it from here?
Breanna
Yes this is what I got, is it correct?
C. 4 with multiplicity 2 ?
C. 4 with multiplicity 2 ?
MathMate
No, it is not the case. There are four roots for a quartic equation, so one single root with multiplicity of 2 does not suffice.
When you have the factor (x+4)<sup>2</sup>, that implies x=-4 with multiplicity of 2.
If you repeat the process with the factor (x-4)<sup>2</sup>, you will find the answer you need.
When you have the factor (x+4)<sup>2</sup>, that implies x=-4 with multiplicity of 2.
If you repeat the process with the factor (x-4)<sup>2</sup>, you will find the answer you need.
Breanna
Wow I am confused now. So does the answer include the i? which is
B. – 4i with multiplicity 2 and 4i with multiplicity 2
OR
A. – 4 with multiplicity 2 and 4 with multiplicity 2
I am goin to say A but I could be wrong.
B. – 4i with multiplicity 2 and 4i with multiplicity 2
OR
A. – 4 with multiplicity 2 and 4 with multiplicity 2
I am goin to say A but I could be wrong.
MathMate
A is correct. The roots are real, so there is no i involved.
There are two distinct roots, ±4 each with multiplicity of 2. So A is the answer.
There are two distinct roots, ±4 each with multiplicity of 2. So A is the answer.