Question

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

Answer 1

Construct the coefficient of each power of `x` in descending order to find:

`(x-2)(2x+3)(3-x) = -2x^3+7x^2+3x-18`

Explanation:

For each power of `x` in descending order, pick out the different combinations of terms, using one from each binomial, such that when multiplied together they will give the target power of `x` and add them together. For brevity, omit the `x`'s as you are multiplying and adding the coefficients...

So:

`color(blue)(x^3)` : `(1*2*-1) = color(red)(-2)`

`color(blue)(x^2)` : `(1*2*3)+(1*3*-1)+(-2*2*-1)`

`= 6-3+4 = color(red)(7)`

`color(blue)(x)` : `(1*3*3)+(-2*2*3)+(-2*3*-1)`

`= 9-12+6 = color(red)(3)`

`color(blue)(1)` : `(-2*3*3) = color(red)(-18)`

Hence `(x-2)(2x+3)(3-x) = -2x^3+7x^2+3x-18`

Check: Try `x=1` ...

`(x-2)(2x+3)(3-x) = (1-2)(2+3)(3-1)`

`= -1*5*2 = color(red)(-10)`

`-2x^3+7x^2+3x-18 = -2+7+3-18 = color(red)(-10)`