Solve 15x+2x^2-x^3

$$15 x + 2 x ^ { 2 } - x ^ { 3 }$$

Factor

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

Solution Steps

Factor out x.

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

Consider 15+2x-x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+2x+15$$

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

$$a+b=2 ab=-15=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$a=5 b=-3$$

Rewrite -x^{2}+2x+15 as \left(-x^{2}+5x\right)+\left(-3x+15\right).

$$\left(-x^{2}+5x\right)+\left(-3x+15\right)$$

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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