Solve x^3-3x^2-33x+35

$$x ^ { 3 } - 3 x ^ { 2 } - 33 x + 35$$

Factor

$$\left(x-7\right)\left(x-1\right)\left(x+5\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 35 and q divides the leading coefficient 1. One such root is 7. Factor the polynomial by dividing it by x-7.

$$\left(x-7\right)\left(x^{2}+4x-5\right)$$

Consider x^{2}+4x-5. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-5\right)=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=5$$

Rewrite x^{2}+4x-5 as \left(x^{2}-x\right)+\left(5x-5\right).

$$\left(x^{2}-x\right)+\left(5x-5\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x-1\right)+5\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$\left(x-7\right)\left(x-1\right)\left(x+5\right)$$

Evaluate

$$\left(x-7\right)\left(x-1\right)\left(x+5\right)$$