Solve 2x^3-7x^2-30x

$$2 x ^ { 3 } - 7 x ^ { 2 } - 30 x$$

Factor

$$x\left(x-6\right)\left(2x+5\right)$$

Solution Steps

Factor out x.

$$x\left(2x^{2}-7x-30\right)$$

Consider 2x^{2}-7x-30. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\left(-30\right)=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -7.

$$a=-12 b=5$$

Rewrite 2x^{2}-7x-30 as \left(2x^{2}-12x\right)+\left(5x-30\right).

$$\left(2x^{2}-12x\right)+\left(5x-30\right)$$

Factor out 2x in the first and 5 in the second group.

$$2x\left(x-6\right)+5\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(2x+5\right)$$

Rewrite the complete factored expression.

$$x\left(x-6\right)\left(2x+5\right)$$

Evaluate

$$x\left(x-6\right)\left(2x+5\right)$$