find all zeros of x^4-5x^3+10^2-20x+24 (leave all irrational answers in radical form)

Question

i simply don't understand this question, i can find the rational value but never can i calculate how to find the irrational values nor the imaginary number values. what mistake am I making ?

Answer 1

I am pretty sure you forgot the x in 10^2 and it should have been x^4-5x^3+10x^2-20x+24

let f(x) = x^4-5x^3+10x^2-20x+24
try x = ±1, ±2, ±3, etc, that is, factors of 24
I found f(2) and f(3) = 0
so (x-2) and (x-3) would be factors.

Doing synthetic division consecutively by x-2 and x-3 gave me
x^4-5x^3+10^2-20x+24
= (x-2)(x-3)(x^2 + 4)

set x^2 + 4 = 0
x^2 = -4
x = ±2i

so zeros are 2, 3, 2i, and -2i

Answer 2

Thank you sooooo much !!!