Solve 2x^3-9x^2+7x+6

$$2 x ^ { 3 } - 9 x ^ { 2 } + 7 x + 6$$

Factor

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

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

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

Consider 2x^{2}-3x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-2\right)=-4$$

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 -4.

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

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

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

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

Factor out 2x in 2x^{2}-4x.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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