Question

How do you multiply `(6z^2 - 4z + 1)(8 - 3z)`?

Answer 1

`-18z^3+60z^2-35z+8`

Explanation:

When multiplying polynomials, as we see here, we must distribute everything.

Every term that is in the trinomial `6z^2-4z+1` must be multiplied individually by both terms in the following binomial `8-3z`. Then, all these multiplied terms will be added to one another to form a large polynomial.

Let's break down what we'll multiply:

`"("underbrace(color(green)(6z^2)underbrace(color(blue)(-4z)+underbrace(color(red)1")("8-3z)_(color(red)(1(8-3z))))_color(blue)(-4z(8-3z)))_color(green)(6z^2(8-3z))")"`

So, we see that the `(8-3z)` term is distributed to each term within `(6z^2-4z+1)`.

Adding these all together, we see that

`(6z^2-4z+1)(8-3z)=color(green)(6z^2(8-3z))+color(blue)((-4z)(8-3z))+color(red)(1(8-3z)`

Distributing each, we obtain

`=color(green)(48z^2-18z^3)+color(blue)(-32z+12z^2)+color(red)(8-3z)`

Now, to simplify, sort this by degree (combine like terms):

`=-18z^3+underbrace(48z^2+12z^2) _ (48+12=60)underbrace(-32z-3z) _ (-32-3=-35)+8`

`=-18z^3+60z^2-35z+8`