Question

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

Hint: Factoring by grouping means that we have to group terms with common factors before factoring. This can be done by grouping a pair of terms and then factor each pair of two terms.

Given expression is ${x^3} - {x^2} - x + 1$

We can write the above polynomial as ${x^2}(x - 1) - 1(x - 1)$
$\Rightarrow (x - 1)({x^2} - 1)$ $\because \left[ {{a^2} - {b^2} = (a + b)(a - b)} \right]$
$\Rightarrow (x - 1)(x - 1)(x + 1)$
$\therefore {x^3} - {x^2} - x + 1$ can be factorized into ${\left( {x - 1} \right)^2}(x + 1)$

Note:
Here we grouped the first two terms together and then the last two terms together. Later we took out the common term from each expression. Then factor out the common binomial.