Chapter 1.3, Problem 88E

To determine

To calculate: The factors of the expression $x^5+x^4+x+1$ .

Answer to Problem 88E

The factors of the expression $x^5+x^4+x+1$ are $\left(x^4+1\right)$ and $\left(x+1\right)$ .

Explanation of Solution

Given information:

The expression $x^5+x^4+x+1$ .

Formula used:

To factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

Calculation:

Consider the given expression $x^5+x^4+x+1$ .

Recall that to factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

Apply it,

  $\begin{array}{c}{x}^5+x^4+x+1=\left(x^5+x^4\right)+\left(x+1\right)\\ =x^4\left(x+1\right)+1\left(x+1\right)\\ =\left(x^4+1\right)\left(x+1\right)\end{array}$

Thus, the factors of expression $x^5+x^4+x+1$ are $\left(x^4+1\right)$ and $\left(x+1\right)$ .