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

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

Factor

$$x\left(x-1\right)\left(2x-7\right)$$

Solution Steps

Factor out x.

$$x\left(2x^{2}-9x+7\right)$$

Consider 2x^{2}-9x+7. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-9 ab=2\times 7=14$$

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 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -9.

$$a=-7 b=-2$$

Rewrite 2x^{2}-9x+7 as \left(2x^{2}-7x\right)+\left(-2x+7\right).

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

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

$$x\left(2x-7\right)-\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$x\left(2x-7\right)\left(x-1\right)$$

Evaluate

$$x\left(x-1\right)\left(2x-7\right)$$