Question

How do you factor the trinomial `x^2-2x+1`?

Answer 1

See below

Explanation:

This is a quadratic and, if all else fails, you revert to the Quadratic Formula .

For generalised quadratic `ax^2 + bx + c = 0`, we know, from the Quadratic Formula, that its roots are:

`x_(1,2) = (-b pmsqrt (b^2 - 4ac))/(2a)`

In this particular case, we get:

`x_(1,2) = (2 pmsqrt (4 - 4))/(2) = 1`, ie repeated roots

Of course we are smarter than (!) that and we will have noted at the outset that `x^2-2x+1` can be factored as:

`x^2-2x+1`

`= (x-1)(x-1)`

Factoring, without the sometimes drudgery of the Quadratic Formula, comes down to experience.

If you see that:

`(x-alpha)(x - beta) = x^2 - (alpha + beta)x + alpha beta`

...then you can start to look for patterns.

In this case, the patterns were:

• `alpha beta = 1`

• `-(alpha + beta) = -2`, ie `(alpha + beta) = 2`.

It follows that `alpha = beta = 1`.