Solve x^5-13x^4-14x^3

$$x ^ { 5 } - 13 x ^ { 4 } - 14 x ^ { 3 }$$

Factor

$$\left(x-14\right)\left(x+1\right)x^{3}$$

Solution Steps

Factor out x^{3}.

$$x^{3}\left(x^{2}-13x-14\right)$$

Consider x^{2}-13x-14. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\left(-14\right)=-14$$

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 -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -13.

$$a=-14 b=1$$

Rewrite x^{2}-13x-14 as \left(x^{2}-14x\right)+\left(x-14\right).

$$\left(x^{2}-14x\right)+\left(x-14\right)$$

Factor out x in x^{2}-14x.

$$x\left(x-14\right)+x-14$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$x^{3}\left(x-14\right)\left(x+1\right)$$

Evaluate

$$\left(x-14\right)\left(x+1\right)x^{3}$$