Solve 2x^3-17x^2+15x

$$2 x ^ { 3 } - 17 x ^ { 2 } + 15 x$$

Factor

$$x\left(x-1\right)\left(2x-15\right)$$

Solution Steps

Factor out x.

$$x\left(2x^{2}-17x+15\right)$$

Consider 2x^{2}-17x+15. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-17 ab=2\times 15=30$$

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 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -17.

$$a=-15 b=-2$$

Rewrite 2x^{2}-17x+15 as \left(2x^{2}-15x\right)+\left(-2x+15\right).

$$\left(2x^{2}-15x\right)+\left(-2x+15\right)$$

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

$$x\left(2x-15\right)-\left(2x-15\right)$$

Factor out common term 2x-15 by using distributive property.

$$\left(2x-15\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$x\left(2x-15\right)\left(x-1\right)$$

Evaluate

$$x\left(x-1\right)\left(2x-15\right)$$