using the factor theorem, find the prime factors of the polynomial?

Question

x^3 + 5x^2 + 2x-8

Thanks!

Answer 1

Note that 1 is a zero of this polynomial, since 1 + 5 + 2-8 = 0.
So by the factor theorem, x-1 is a factor.
By long division or synthetic division, the quotient
is x² + 6x + 8.
So the factors of the polynomial are
x -1, x + 2 and x + 4.

Answer 2

f(x) = x^3 + 5x^2 + 2x - 8

f(1) = 1 + 5 + 2 - 8 = 0

x = 1 is root and (x - 1) is factor of f(x)

f(-2) = -8 + 20 - 4 - 8 = 0

so x = -2 is a another root and x + 2 is a factor

f(-4) = (-4)^3 + 5(-4)^2 + 2(-4) - 8

f(-4) = -64 +80 - 8 - 8 = 0

so x = -4 is also a root and (x + 4) is another factor

All the factors of f(x) are (x -1), (x + 2) and (x+4)