Question

How do you simplify `(x + 6) ( x + 1) ( x + 4)`?

Answer 1

`(x+6)(x+1)(x+4) = x^3+11x^2+34x+24`

Explanation:

Note that:

`(x+a)(x+b)(x+c)`

`= x^3+(a+b+c)x^2+(ab+bc+ca)x+abc`

The polynomials:

`{ (a+b+c), (ab+bc+ca), (abc) :}`

are called the elementary symmetric polynomials in `a`, `b` and `c`. Interestingly, any symmetric polynomial in `a`, `b` and `c` is expressible in terms of these three elementary ones.

Whatever they're called, notice the ways in which we need to collect the coefficients to multiply the `3` binomials.

Hence we find:

`(x+6)(x+1)(x+4) = x^3+(6+1+4)x^2+(6*1+1*4+4*6)x+6*1*4`

`color(white)((x+6)(x+1)(x+4)) = x^3+11x^2+34x+24`