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

Apr 14, 2016

$- 18 {z}^{3} + 60 {z}^{2} - 35 z + 8$

#### Explanation:

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

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

Let's break down what we'll multiply:

$\text{("underbrace(color(green)(6z^2)underbrace(color(blue)(-4z)+underbrace(color(red)1")("8-3z)_(color(red)(1(8-3z))))_color(blue)(-4z(8-3z)))_color(green)(6z^2(8-3z))")}$

So, we see that the $\left(8 - 3 z\right)$ term is distributed to each term within $\left(6 {z}^{2} - 4 z + 1\right)$.

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

$= \textcolor{g r e e n}{48 {z}^{2} - 18 {z}^{3}} + \textcolor{b l u e}{- 32 z + 12 {z}^{2}} + \textcolor{red}{8 - 3 z}$

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

$= - 18 {z}^{3} + {\underbrace{48 {z}^{2} + 12 {z}^{2}}}_{48 + 12 = 60} {\underbrace{- 32 z - 3 z}}_{- 32 - 3 = - 35} + 8$

$= - 18 {z}^{3} + 60 {z}^{2} - 35 z + 8$