Posted by Breanna on .
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

Algebra 
MathMate,
f(x) = (x^{2}16)^{2}
= ((x+4)(x4))^{2}
= (x+4)^{2}(x4)^{2}
Can you take it from here? 
Algebra 
Breanna,
Yes this is what I got, is it correct?
C. 4 with multiplicity 2 ? 
Algebra 
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)^{2}, that implies x=4 with multiplicity of 2.
If you repeat the process with the factor (x4)^{2}, you will find the answer you need. 
Algebra 
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. 
Algebra 
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.