Question

Given that -2i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem if applicable.

Answer 1

So, the completely factored form of the given polynomial is:
`f(x)=\left(x^2+4\right)(x-6)(x+1)`

The conjugate root theorem states that if
`$a+b i$` is a root of a polynomial with real coefficients, then its conjugate
`$a-b i$` is also a root of the polynomial.

We can start by factoring
`$(x-6)$` out of the polynomial:

`x^4-5 x^3-2 x^2-20 x-24=(x-6)\left(x^3+x^2+4 x+4\right)`

The remaining polynomial,
`$x^3+x^2+4 x+4$`, can be factored further using grouping:

`x^3+x^2+4 x+4=\left(x^3+x^2\right)+(4 x+4)=x\left(x^2+1\right)+4(x+1)=(x+1)\left(x^2+4\right)`

Now, we can combine the factors we found to get the complete factorization of the original polynomial:

`(x-6)(x+1)\left(x^2+4\right)`

Complete Question:
Given that
`$-2 i$` is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable:

`f(x)=x^4-5 x^3-2 x^2-20 x-24`