Question

use the rational zero theorem to find all the zeros of the polynomial function. Use the factor f over the real numbers.

f(x)=x^4+2x^3-7x^2-8x+12

x=

f(x)=

Answer 1

try factors of 12
on the first try, I got f(1) = 0
so x-1 is a factor

by long algebraic or by synthetic division , I got
x^4+2x^3-7x^2-8x+12
= (x-1)(x^3 + 3x^2 - 4x - 12)

grouping the x^3 + 3x^2 - 4x - 12
= x^2(x+3) - 4(x+3)
= (x+3)(x^2-4)
= (x+3)(x+2)(x-2)

so
x^4+2x^3-7x^2-8x+12
= (x-1)(x-2)(x+2)(x+3)

(had I been patient, I would have found
f(2) = 0, f(-3)=0 and f(-2) = 0 as well)

the zeros are
1, 2, -2, -3