Question

How do you multiply `(x-1)(x-2)(x-3)`?

Answer 1

`(x-1)(x-2)(x-3) = x^3-6x^2+11x-6`

Explanation:

It is helpful to know that:

`(x-alpha)(x-beta)(x-gamma)`

`= x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma`

In this identity the expressions forming the coefficients of `x^k` are (apart from the alternating signs), the elementary symmetric polynomials in `alpha, beta, gamma`, namely:

• `alpha+beta+gamma`

• `alphabeta+betagamma+gammaalpha`

• `alphabetagamma`

With `alpha=1`, `beta=2` and `gamma=3`, we find:

`alpha+beta+gamma=1+2+3=6`

`alphabeta+betagamma+gammaalpha=(1 * 2) + (2 * 3) + (3 * 1) = 2+6+3=11`

`alphabetagamma = 1 * 2 * 3 = 6`

So:

`(x-1)(x-2)(x-3) = x^3-6x^2+11x-6`