Solve x^7-17x^6-18x^5

$$x ^ { 7 } - 17 x ^ { 6 } - 18 x ^ { 5 }$$

Factor

$$\left(x-18\right)\left(x+1\right)x^{5}$$

Solution Steps

Factor out x^{5}.

$$x^{5}\left(x^{2}-17x-18\right)$$

Consider x^{2}-17x-18. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=-17 ab=1\left(-18\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -17.

$$a=-18 b=1$$

Rewrite x^{2}-17x-18 as \left(x^{2}-18x\right)+\left(x-18\right).

$$\left(x^{2}-18x\right)+\left(x-18\right)$$

Factor out x in x^{2}-18x.

$$x\left(x-18\right)+x-18$$

Factor out common term x-18 by using distributive property.

$$\left(x-18\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$x^{5}\left(x-18\right)\left(x+1\right)$$

Evaluate

$$\left(x-18\right)\left(x+1\right)x^{5}$$