Solve 6x^3-11x^2+x+4

$$6 x ^ { 3 } - 11 x ^ { 2 } + x + 4$$

Factor

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

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

$$\left(x-1\right)\left(6x^{2}-5x-4\right)$$

Consider 6x^{2}-5x-4. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-4\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$a=-8 b=3$$

Rewrite 6x^{2}-5x-4 as \left(6x^{2}-8x\right)+\left(3x-4\right).

$$\left(6x^{2}-8x\right)+\left(3x-4\right)$$

Factor out 2x in 6x^{2}-8x.

$$2x\left(3x-4\right)+3x-4$$

Factor out common term 3x-4 by using distributive property.

$$\left(3x-4\right)\left(2x+1\right)$$

Rewrite the complete factored expression.

$$\left(3x-4\right)\left(x-1\right)\left(2x+1\right)$$

Evaluate

$$6x^{3}-11x^{2}+x+4$$