Question

I am having touble understanding:

how to factor the polynomial :
8x^4 -6x^3 -57x^2 -6x-65, into its linear factors, given that i is a zero.

I tried to use synthetic division, but i keep getting a remainder... please help me understand how to solve it.

Answer 1

you are given that x = i is a solution
Complex roots, just like irrational roots, always come in conjugate pairs, so x = -i is another solution
so you know
(x+i) and (x-i) are factors
(x+1)(x-1) = x^2 + 1
which means x^2 + 1 is factor of your given polynomial

now do a long division of 8x^4 -6x^3 -57x^2 -6x-65 by (x^2 + 1)
I am not going to show this here, but you should get
8x^4 -6x^3 -57x^2 -6x-65
= (x^1 + 1)(8x^2 - 6x - 65)

I trust you will be able to further factor the last part, it does factor