Given that 4−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4−2x3−23x2+248x−224

Question

Given that 4−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

f(x)=x^4−2x^3−23x^2+248x−224

Answer 1

If 4-4i is a zero, then is its conjugate, 4+4i, as well (I think you know why?).

Next 4-4i and 4+4i are zeros, this means that 

x⁴-2x³-23x²+248x-224=(x-4+4i)(x-4-4i)g(x), (A)

where g(x) is a polinomial od degree 2.

Since (x-4+4i)(x-4-4i)=x²-8x+32,(A) is equaivalent to 

x⁴-2x³-23x²+248x-224=(x²-8x+32)g(x) (B)

Now dividing LHS of (B) by x²-8x+32 you find that g(x)=x²+6x-7.

So

x⁴-2x³-23x²+248x-224=(x²-8x+32)(x²+6x-7)  (C)

 

It is easy to see that x²+6x-7=(x-1)(x+7).

Finally, you get

x⁴-2x³-23x²+248x-224=(x-1)(x+7)(x-4+4i)(x-4-4i)