Question

How do you factor $x^3+x^2+x+1$ by grouping?

Answer 1

Factorize using the grouping method

The given expression is $x^3+x^2+x+1$.

Follow the below steps to factorize an equation of the above form:

• Group two pairs of terms.
• Factor out the common terms from each group.
• Factor out the common group so obtained.

Then, the given expression can be factorized as follows,

$x^3+x^2+x+1=\left(x^3+x^2\right)+\left(x+1\right)$

$\Rightarrow x^3+x^2+x+1=x^2\left(x+1\right)+1\left(x+1\right)$ $\mathrm{[}\mathrm{\because }\mathrm{}\mathrm{ab}\mathrm{+}\mathrm{ac}\mathrm{=}\mathrm{a}\left(b+c\right)\mathrm{]}$

$\Rightarrow x^3+x^2+x+1=\left(x+1\right)\left(x^2+1\right)$

Hence, the factors of $x^3+x^2+x+1$ obtained by using the common factor method is $\left(x+1\right)\left(x^2+1\right)$.