Solve 1+x+x^2-111

$$1 + x + x ^ { 2 } - 111$$

Factor

$$\left(x-10\right)\left(x+11\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{2}+x-110$$

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-110. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-110\right)=-110$$

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 -110.

$$-1,110 -2,55 -5,22 -10,11$$

Calculate the sum for each pair.

$$-1+110=109 -2+55=53 -5+22=17 -10+11=1$$

The solution is the pair that gives sum 1.

$$a=-10 b=11$$

Rewrite x^{2}+x-110 as \left(x^{2}-10x\right)+\left(11x-110\right).

$$\left(x^{2}-10x\right)+\left(11x-110\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x-10\right)+11\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x+11\right)$$

Evaluate

$$\left(x-10\right)\left(x+11\right)$$

Solution Steps

Subtract 111 from 1 to get -110.

$$-110+x+x^{2}$$