Solve 15x^3-4x^2-19x

$$15 x ^ { 3 } - 4 x ^ { 2 } - 19 x$$

Factor

$$x\left(15x-19\right)\left(x+1\right)$$

Solution Steps

Factor out x.

$$x\left(15x^{2}-4x-19\right)$$

Consider 15x^{2}-4x-19. Factor the expression by grouping. First, the expression needs to be rewritten as 15x^{2}+ax+bx-19. To find a and b, set up a system to be solved.

$$a+b=-4 ab=15\left(-19\right)=-285$$

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 -285.

$$1,-285 3,-95 5,-57 15,-19$$

Calculate the sum for each pair.

$$1-285=-284 3-95=-92 5-57=-52 15-19=-4$$

The solution is the pair that gives sum -4.

$$a=-19 b=15$$

Rewrite 15x^{2}-4x-19 as \left(15x^{2}-19x\right)+\left(15x-19\right).

$$\left(15x^{2}-19x\right)+\left(15x-19\right)$$

Factor out x in 15x^{2}-19x.

$$x\left(15x-19\right)+15x-19$$

Factor out common term 15x-19 by using distributive property.

$$\left(15x-19\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$x\left(15x-19\right)\left(x+1\right)$$

Evaluate

$$x\left(15x-19\right)\left(x+1\right)$$