How do you factor $x^{2} - 2 x + 1$ $x^2-2x+1$ ?

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How do you factor $x^{2} - 2 x + 1$ $x^2-2x+1$ ?

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Answer 1 · 2015-07-25T19:13:50

This is a perfect square trinomial:

$x^{2} - 2 x + 1 = (x - 1) (x - 1) = {(x - 1)}^{2}$ $x^2-2x+1 = (x-1)(x-1) = (x-1)^2$

Explanation:

In general, you can recognise these perfect square trinomials, because they take the form $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ $a^2+2ab+b^2 = (a+b)^2$ or $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ $a^2-2ab+b^2 = (a-b)^2$

In this particular case, there's a little trick to help spot it:

The coefficients of your quadratic - ignoring the signs - are $1$ $1$ , $2$ $2$ and $1$ $1$ . Does the sequence $121$ $1 2 1$ ring any bells? Well $121 = 11 \cdot 11 = 11^{2}$ $121 = 11*11 = 11^2$ and $x^{2} - 2 x + 1 = {(x - 1)}^{2}$ $x^2-2x+1 = (x-1)^2$ , where the coefficients of $x - 1$ $x-1$ - ignoring signs - are $1$ $1$ and $1$ $1$ .

This works with a few other trinomials, like:

$x^{2} + 6 x + 9 = {(x + 3)}^{2}$ $x^2+6x+9 = (x+3)^2$ like $169 = 13^{2}$ $169 = 13^2$

It only works if the numbers are small enough, but it can help.

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How do you factor $x^{2} - 2 x + 1$ $x^2-2x+1$ ?

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