Solve 2x^3-13x^2+17x+12

$$2 x ^ { 3 } - 13 x ^ { 2 } + 17 x + 12$$

Factor

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

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 12 and q divides the leading coefficient 2. One such root is 3. Factor the polynomial by dividing it by x-3.

$$\left(x-3\right)\left(2x^{2}-7x-4\right)$$

Consider 2x^{2}-7x-4. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\left(-4\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -7.

$$a=-8 b=1$$

Rewrite 2x^{2}-7x-4 as \left(2x^{2}-8x\right)+\left(x-4\right).

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

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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