X^3 + 5x^2 - 8x^2 - 40x + 7x + 35

Remark

x - 1 is a root.

Proof

x - 1 = 0

x = 1

(1) ^3 + 5 (1) ^2 - 8 (1) ^2 - 40 (1) + 7 (1) + 35

1 + 5 - 8 - 40 + 7 + 35

6 + 7 + 35 - 48 = 0

Solution

Divide x - 1 into the original equation.

x^3 + 5x^2 - 8x^2 - 33x + 35

x - 1 || x^3 - 3x^2 - 33x + 35 || x^2 - 2x - 35

x^3 - x^2

-2x^2 - 33x

-2x^2 + 2x

- 35x + 35

-35x + 35

0

x^2 - 2x - 35 factors into (x - 7) (x + 5)

Complete factorization

(x - 1) (x - 7) (x + 5)