## 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:

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

Show Answer
Use the distributive property to multiple the polynomials. Multiple the terms \(2x\) and \(-18\) by the second polynomial:

\(2x \cdot (3x + 3) - 18 \cdot (3x + 3)\)

\((6x^2+6x) + (-54x-54)\)

Now simplify by collecting like terms:

\(6x^2-48x-54\)

\((2x+1)(3x^2−x+4) \)

Show Answer
Use the distributive property to multiple the polynomials. Multiple the terms \(2x\) and \(1\) by the second polynomial:

\(2x \cdot (3x^2 - x +4) + 1\cdot(3x^2 - x +4)\)

\((6x^3-2x^2+8x)+(2x^2-x+4)\)

Now simplify by collecting like terms:

\(6x^3+x^2+7x+4\)