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Question

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

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Solution

Factorize using the grouping methodThe 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)$$⇒{x}^{3}+{x}^{2}+x+1={x}^{2}\left(x+1\right)+1\left(x+1\right)$ $\mathrm{\left[}\mathrm{\because }\mathrm{}\mathrm{ab}\mathrm{+}\mathrm{ac}\mathrm{=}\mathrm{a}\left(b+c\right)\mathrm{\right]}$$⇒{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 $\left(x+1\right)\left({x}^{2}+1\right)$.

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