SOLUTION: Hi I'm having trouble understanding. Finding all zero's of a polynomial function given one zero. My problem is: f(x)= x^4-32x^2-144; one zero is -2i The answer given is, 2i,6,-6

Algebra ->  Test  -> Lessons -> SOLUTION: Hi I'm having trouble understanding. Finding all zero's of a polynomial function given one zero. My problem is: f(x)= x^4-32x^2-144; one zero is -2i The answer given is, 2i,6,-6       Log On


   

Question 670349: Hi I'm having trouble understanding. Finding all zero's of a polynomial function given one zero. My problem is: f(x)= x^4-32x^2-144; one zero is -2i
The answer given is, 2i,6,-6 but I'm not understanding how. Thank you!
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
To find the zeros of this polynomial function we need to set f%28x%29 equal to 0 and factor the equation.
....since you are given one zero which is -%282i%29, it is result of the square of negative number that could be only sqrt%28-4%29; only sqrt%28-4%29 gives you root -2i, as 2i as well
therefore, you are actually given two roots, and now we will find remaining two
0=+x%5E4-32x%5E2-144.....replace -32x%5E2 with -36x%5E2%2B4x%5E2
x%5E4-36%5E2%2B4x%5E2-144.....group
%28x%5E4-36%5E2%29%2B%284x%5E2-144%29...factor out x%5E2 from the first group and 4 from the second group
x%5E2%28x%5E2-36%29%2B4%28x%5E2-36%29
%28x%5E2%2B4%29%28x%5E2-36%29
%28x%5E2%2B4%29%28x-6%29%28x%2B6%29

Now we set each factor to zero:
x%5E2%2B4=0...............x%5E2=-4 .........=> given roots: the imaginary numbers -2i and 2i
x-6=0........x=6..................real root
x%2B6=0........x=-6..............real root

show it on a graph:

+graph%28+600%2C+600%2C+-10%2C10%2C+-420%2C+200%2C%28x%5E2%2B4%29%28x-6%29%28x%2B6%29%29+