Solve g(x)=6x^3-x^2-5x+2

$$g ( x ) = 6 x ^ { 3 } - x ^ { 2 } - 5 x + 2$$

Factor

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

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 2 and q divides the leading coefficient 6. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(6x^{2}-7x+2\right)$$

Consider 6x^{2}-7x+2. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-7 ab=6\times 2=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

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

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-4 b=-3$$

Rewrite 6x^{2}-7x+2 as \left(6x^{2}-4x\right)+\left(-3x+2\right).

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

Factor out 2x in the first and -1 in the second group.

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

Factor out common term 3x-2 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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