Question

How do you multiply `(4x-5)(2x^5 + x3 - 1)`?

Answer 1

For each power of `x` from `x^6` down to `x^0`, pick out the pairs that multiply to give that power and add them to find:

`(4x-5)(2x^5+x^3-1)`

`= 8x^6-10x^5+4x^4-5x^3-4x+5`

Explanation:

Looking at each power of `x` in turn from `x^6` down to `x^0`, pick out the pairs that multiply to give a term with that power of `x`:

`x^6` : `4x*2x^5 = 8x^6`

`x^5` : `-5*2x^5 = -10x^5`

`x^4` : `4x*x^3 = 4x^4`

`x^3` : `-5 * x^3 = -5x^3`

`x^2` : none

`x` : `4x*-1 = -4x`

`1` : `-5*-1 = 5`

Add to get:

`8x^6-10x^5+4x^4-5x^3-4x+5`

Normally when multiplying two polynomials in this way you would have two pairs to multiply and add for most of the powers of `x`, but a similar approach works.