Factorise the expression `x-1 - (x-1)^2`
Follow the steps 1)-4)
1) First expand out the bracketed term:
`x-1 -(x-1)^2 = x- 1 - (x-1)(x-1) `
`= x - 1 - (x^2 - x - x + 1)`
`= x - 1 - (x^2 - 2x + 1)`
2) Now gather terms:
`x - 1 - (x-1)^2 = x - 1 - x^2 + 2x - 1 = -x^2 + 3x - 2`
3) Now, to factorise the original expression, we are looking for constants `a` and `b` such that
`-x^2 + 3x - 2 = (-x+a)(x+b)`
4) From this we can see that `(a-b) = 3` and `ab = -2`.
The idea is to spot the simple solution `a=2` and `b=-1`. This can be logically arrived at by noting that `pm 1` and `pm 2` are the only integer possibilities if `ab= -2`and that the only way these possible values of `a` and `b` can give `a-b=3` is by combining a positive 2 and a negative 1.
The expression x - 1 - (x-1)^2 factorises to (2-x)(x-1).
Alternatively, factor out an `x-1` right away.
distribute the exponent
distribute the negative sign
combine like terms
put it in factored form
factors of 2 that minus to 1 = 2 and 1
plug them in
factor it out