Solve 30x^3-5x^2-60x?

$$30 x ^ { 3 } - 5 x ^ { 2 } - 60 x ?$$

Factor

$$5x\left(2x-3\right)\left(3x+4\right)$$

Solution Steps

Factor out 5.

$$5\left(6x^{3}-x^{2}-12x\right)$$

Consider 6x^{3}-x^{2}-12x. Factor out x.

$$x\left(6x^{2}-x-12\right)$$

Consider 6x^{2}-x-12. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-1 ab=6\left(-12\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -1.

$$a=-9 b=8$$

Rewrite 6x^{2}-x-12 as \left(6x^{2}-9x\right)+\left(8x-12\right).

$$\left(6x^{2}-9x\right)+\left(8x-12\right)$$

Factor out 3x in the first and 4 in the second group.

$$3x\left(2x-3\right)+4\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(3x+4\right)$$

Rewrite the complete factored expression.

$$5x\left(2x-3\right)\left(3x+4\right)$$

Evaluate

$$5x\left(2x-3\right)\left(3x+4\right)$$