Solve 2x^3-9x^2+10x

$$2 x ^ { 3 } - 9 x ^ { 2 } + 10 x$$

Factor

$$x\left(x-2\right)\left(2x-5\right)$$

Solution Steps

Factor out x.

$$x\left(2x^{2}-9x+10\right)$$

Consider 2x^{2}-9x+10. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=-9 ab=2\times 10=20$$

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 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -9.

$$a=-5 b=-4$$

Rewrite 2x^{2}-9x+10 as \left(2x^{2}-5x\right)+\left(-4x+10\right).

$$\left(2x^{2}-5x\right)+\left(-4x+10\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(2x-5\right)-2\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(x-2\right)$$

Rewrite the complete factored expression.

$$x\left(2x-5\right)\left(x-2\right)$$

Evaluate

$$x\left(x-2\right)\left(2x-5\right)$$