Solve 5x^3-65x^2+150x

$$5 x ^ { 3 } - 65 x ^ { 2 } + 150 x$$

Factor

$$5x\left(x-10\right)\left(x-3\right)$$

Solution Steps

Factor out 5.

$$5\left(x^{3}-13x^{2}+30x\right)$$

Consider x^{3}-13x^{2}+30x. Factor out x.

$$x\left(x^{2}-13x+30\right)$$

Consider x^{2}-13x+30. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\times 30=30$$

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 -13.

$$a=-10 b=-3$$

Rewrite x^{2}-13x+30 as \left(x^{2}-10x\right)+\left(-3x+30\right).

$$\left(x^{2}-10x\right)+\left(-3x+30\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(x-10\right)-3\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x-3\right)$$

Rewrite the complete factored expression.

$$5x\left(x-10\right)\left(x-3\right)$$

Evaluate

$$5x\left(x-10\right)\left(x-3\right)$$