# Expressions and the Distributive Property

## Key Questions

See examples below

#### Explanation:

Whatever is outside of the parenthesis, we must multiply it by all terms on the inside.

Example: $11 \left(3 x + 9 y\right)$

In this case, we would multiply the $11$ by both of the terms in the parenthesis to get

$33 x + 99 y$

What if we have two sets of parenthesis?

$\left(2 x + 8\right) \left(3 x + 11\right)$

We multiply every term in the first parenthesis by everyone in the second. We are essentially doing the distributive property twice.

This method is sometimes called FOIL, standing for Firsts, Outsides, Insides, Lasts. This is the order we multiply in. Going back to our example

$\left(2 x + 8\right) \left(3 x + 11\right)$

• We multiply the first terms: $2 x \cdot 3 x = \textcolor{\lim e}{6 {x}^{2}}$
• Outside terms: $2 x \cdot 11 = \textcolor{\lim e}{22 x}$
• Inside terms: $8 \cdot 3 x = \textcolor{\lim e}{24 x}$
• Last terms: $8 \cdot 11 = \textcolor{\lim e}{88}$

Now we have

$6 {x}^{2} + 22 x + 24 x + 88$ which can be simplified to

$6 {x}^{2} + 46 x + 88$

Hope this helps!

• The distributive property says that $a \left(b + c\right) = a \cdot b + a \cdot c$

Without this you wouldn't be able the expand expressions like:

$\left(x + 1\right) \left(2 x - 4\right)$ into

$x \left(2 x - 4\right) + 1 \left(2 x - 4\right)$, then

$x \cdot 2 x + x \cdot \left(- 4\right) + 1 \cdot 2 x + 1 \cdot \left(- 4\right)$ and then

$2 {x}^{2} - 4 x + 2 x - 4 = 2 {x}^{2} - 2 x - 4$

In other words, you would not be able to 'clear the brackets'

See examples below

#### Explanation:

Distributive property is $a \left(b + c\right) = a b + a c$ and also $\left(a + b\right) \left(c + d\right) = a c + a d + b c + b d$

Imagine you want calculate 7·25

In this case you can say $7 \left(20 + 5\right) = 140 + 35 = 175$

Another one: 23·42=(20+3)·(40+2)=20·40+20·2+3·40+2·3=800+40+120+6=966

see below

#### Explanation:

Let's think about matrices. $A B \ne B A$

Left distribution

$A \left(B + C\right) = A B + A C$

$2 \left(B + C\right) = 2 B + 2 C$

Right distribution

$\left(A + B\right) C = A C + B C$

$\left(A + B\right) \cdot 2 = A \cdot 2 + B \cdot 2$