Question

Factor the expression by grouping terms

$x^5+x^4+x+1$

Short answer

The solution$x^5+x^4+x+1=\left(x^4+1\right)\left(x+1\right)$

Step-by-step solution

Step 1. Apply the concept of distributive property.

If A, B and C are real numbers, then

$A\left(B+C\right)=\left(B+C\right)A=AB+AC$

Step 2. Analyze the expression.

Given$x^5+x^4+x+1$

Let us consider first two terms,$x^5,x^4$

1, 1 have greatest common factor 1

$x^5,x^4$have greatest common factor$x^4$

So, the greatest common factor of the two terms is$x^4$

Step 3. Factor the polynomial.

Now, using the distributive property the polynomial can be factored as:

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