The Distributive Property

The distributive property allows us to take an expression like $$3\left( {4x - 6} \right)$$ and distribute the 3. In other words, we can multiply each term inside the parenthesis by 3 to get:  $$3\left( {4x - 6} \right) = 12x - 18$$. But, why does this work?

Consider the purely numeric example: $$2\left( {2 + 2 + 3} \right)$$
 
Now, instead of thinking of these as numbers, imagine that we have 2 dots, 2 more dots, and 3 more dots, and we want to double the number of dots we have:

$$2\left( { \bullet  \bullet  +  \bullet  \bullet  +  \bullet  \bullet  \bullet } \right)$$

Here, we can either take the total number of dots and multiply them by 2 to get:

$$2 \cdot \left( { \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet } \right) =  \underbrace{\bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet }_{\rm{14 \;dots}}$$

OR:

We can take each group of set of dots we had to begin with and double each smaller set:

$$2\left( { \bullet  \bullet } \right) + 2\left( { \bullet  \bullet } \right) + 2\left( { \bullet  \bullet  \bullet } \right) =  \bullet  \bullet  \bullet  \bullet  +  \bullet  \bullet  \bullet  \bullet  +  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  =  \underbrace{\bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet  \bullet}_{\rm{14 \;dots}}$$

In either case, we end up with 14 dots. In the first case, we add up our original sets of dots first to get a set of 7 dots that we double for our total. In the second case, we distribute the 2 by doubling each of our smaller sets first and then adding them up. Both methods are equivalent.

Example 1: Distributing a constant

Here is another worked example of the distributive property of a constant (a number without a variable) distributed over a polynomial.

Use the distributive property to remove parentheses and then simplify by combining like terms:

$$7(4x+3y-6)+{\large\frac{1}{8}}(3y-8x-7)$$

Follow these steps to distribute and simplify:

First, distribute the 7 over the first set of parentheses by multiplying each term within the parentheses by 7:

 \[\eqalign{&7(4x+3y-6)+{\frac{1}{8}}(3y-8x-7)\\=&28x+21y-42+{\frac{1}{8}}(3y-8x-7)}\]

Now, distribute the $${\frac{1}{8}}$$ over the second set of parentheses by multiplying each term in the parentheses by $${\frac{1}{8}}$$:

\[\eqalign{&28x+21y-42+{\frac{1}{8}}(3y-8x-7)\\=&28x+21y-42+{\frac{3}{8}}y-x-{\frac{7}{8}}}\]

Finally, combine like terms to get the final simplified expression:

\[\eqalign{&28x+21y-42+{\frac{3}{8}}y-x-{\frac{7}{8}}\\ =&27y+{\frac{171}{8}}y-{\frac{343}{8}}}\]

Check your skills by completing the following problems:

1. Distribute the constant over the polynomial:  $$4\left( {5x - 3x^2  + 7} \right)$$

Think about it for a moment and then access this link to view answer.

2. Distribute the constant over the polynomial:   $${\large{1 \over 3}}\left( {63x^3  + 14x^2  + 9x - 5} \right)$$

Think about it for a moment and then access this link to view answer.

3. Use the distributive property, then simplify like terms by combining : $$4\left( {3x + 2y - 9x^2 } \right) + 3\left( {2x + 7x^2  + y} \right)$$

Think about it for a moment and then access this link to view answer.

4. Use the distributive property, then simplify like terms by combining:

$$6\left( {y^2  - 3y + 7} \right) - (4y^2  + 8y - 2)$$

Think about it for a moment and then access this link to view answer.

5. Use the distributive property, then simplify like terms by combining:

$${\large{1 \over 4}}\left( {9x - 8x^2  + 2} \right) - 3\left( {{\large{1 \over 4}}x - 7 + 2x^2 } \right)$$

Think about it for a moment and then access this link to view answer.