Using the given zero, find all other zeros of f(x).
-2i is a zero of f(x)=x^4-21x^2-100

Question

Using the given zero, find all other zeros of f(x).
-2i is a zero of f(x)=x^4-21x^2-100

anonymous · Answer

don't imaginary zeros always come in conjugate pairs?  If so, then another zero would be 2i.

anonymous · Answer

$$Substitute x^2 with T so that the equation is T^2 - 21T - 100 = 0$$

anonymous · Answer

@Ajk I know that, but then there are the other zeros.

anonymous · Answer

maybe once you "un-zero" (i.e. make back into factors), you can foil them to get a quadratic, then divide THAT from the original, which will also leave you with a quadratic, which you can then just use the quadratic formula for the last two.

anonymous · Answer

Okay, First we know that if -2i is a 0 then 2i is also a zero (by conjugate pairs). The coefficent of x^4 is 1 which makes life easier aswell. So we know that two of the roots of this equation are 2i and -2i. From there we can do the following $$x^4 - 21x^2 -100 = (x-2i)(x+2i)(x-a)(x-b)$$ Where a and b are the other two roots, assuming it factors nicely. Expand out the (x-2i)(x+2i) and we get x^2 +4. Now we can do algebraic division, $$\frac{ x^4 -21x^2 -100 }{ x^2+4 }$$ Which gives us $$x^2-25$$Which is of course very nice as it is (x-5)(x+5). So overally we have shown that $$x^4 - 21x^2 -100=(x-2i)(x+2i)(x-5)(x+5)$$So the other zero's are 5,-5 and 2i

anonymous · Answer

Thank you!