# Distributive Property

## Distributive Property

The distributive property is used to multiply polynomials. First let's start with just one number to distribute:

$$4 (x^2 - 3y)$$

Here, the $$4$$ should be multiplied to every term within the brackets:

$$(4)(x^2) + (4)(- 3y)$$

Notice we used an addition sign between the new terms. Remember your integer rules for multipliation (same sign = positive, opposite sign = negative).

$$4x^2 - 12y$$

To multiply polynomials, we need to multipy combinations of all the terms. You can use the acrynom FOIL to remember how to multiply two binomials. FOIL stands for first, outer, inner, last:

$$(3-x^3)(4y+3)$$

 $$(\colorbox{yellow}{3}-x^3)(\colorbox{yellow}{4y}+3)$$ $$(\colorbox{yellow}{3}-x^3)(4y+\colorbox{yellow}{3})$$ $$(3\colorbox{yellow}{-x^3})(\colorbox{yellow}{4y}+3)$$ $$(3\colorbox{yellow}{-x^3})(4y+\colorbox{yellow}{3})$$ $$12y$$ $$9$$ $$-4x^3y$$ $$-3x^3$$ F O I L

Then, add up the terms and collect like terms (if possible):

$$12y + 9 -4x^3y -3x^3$$

Another way to approach the problem is to split up the first polynomial and multiply each term by the second polynomial:

$$(3-x^3)(4y+3)$$

$$(3)(4y+3) + (-x^3)(4y+3)$$

Now distribute just one number:

$$(12y)+(9) + (-4x^3y)+(-3x^3)$$

$$12y + 9 -4x^3y -3x^3$$

We get the same answer as before.

Simplify the following:

$$(2x-18)(3x+3)$$

$$(2x+1)(3x^2−x+4)$$