Question

How do you simplify `(x^2-5x+4) /(x-1)`?

Answer 1

`x-4`.

Explanation:

Find the roots of the numerator: since it is a quadratic formula `ax^2+bx+c` with `a=1`, you can use the sum and product formula: you can write your expression as `x^2-sx+p`, where `s` is the sum of the roots, and `p` is their product. So, we're looking for two numbers `x_0` and `x_1` such that `x_0+x_1=5`, and `x_0x_1=4`. These numbers are easily found to be `1` and `4`.

So, we can write `x^2-sx+p=(x-x_0)(x-x_1)`, and thus

`x^2-5x+4 = (x-1)(x-4)`. Plugging this into the fraction gives

`{cancel((x-1))(x-4)}/{cancel(x-1)`, and the expression simplifies into `x-4`.