Solve 6x^3-5x^2-17x+6

$$6 { x }^{ 3 } -5 { x }^{ 2 } -17x+6$$

Factor

$$2\left(3x-1\right)\left(2x+3\right)\left(\frac{x}{2}-1\right)$$

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 6 and q divides the leading coefficient 6. One such root is \frac{1}{3}. Factor the polynomial by dividing it by 3x-1.

$$\left(3x-1\right)\left(2x^{2}-x-6\right)$$

Consider 2x^{2}-x-6. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-1 ab=2\left(-6\right)=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=-4 b=3$$

Rewrite 2x^{2}-x-6 as \left(2x^{2}-4x\right)+\left(3x-6\right).

$$\left(2x^{2}-4x\right)+\left(3x-6\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-2\right)+3\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(2x+3\right)$$

Rewrite the complete factored expression.

$$\left(x-2\right)\left(3x-1\right)\left(2x+3\right)$$

Evaluate

$$6x^{3}-5x^{2}-17x+6$$