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What is the Distributive Property?

The formalized definition of the distributive property of multiplication over addition is...

\(x(a+b+c+d+...) = xa+xb+xc+xd+...\)

Basically, it tells us that when we have something multiplied by a set of terms inside parentheses, we can get rid of the parentheses by multiplying that thing by each term individually. 

The distributive property can be applied to variables or numbers and it can also be reversed.

Distributive Property with Numbers

When you have only numbers (no variables) inside of the parentheses, it's usually easiest to simplify the expression using order of operations instead of the distributive property.

So, you probably won't ever use the distributive property to simplify expressions with only numbers, although you could if you wanted to.

However, it can be helpful to see an "only numbers" example when you are first learning about the distributive property, because it shows WHY the property works. 

\(3(2+5-1) = 3(2)+3(5)+3(-1)\)

We can see that the distributive property is true because if we use theorder of operations to simplify the left hand side, we get 18.

And if we simplify the right hand side, we also get 18. 

Left Side

\(3(2+5-1)\)

\(3(6)\)

\(18\)

Right Side

\(3(2)+3(5)+3(-1)\)

\(6+15-3\)

\(18\)

These expressions are equivalent because they represent the same amount. They are just written in different ways. 

Distributive Property with Variables

When you are simplifying expressions with variables inside the parentheses, order of operations usually doesn't work because you can't add or subtract unlike terms.

The distributive property allows us to get rid of the parentheses without combining the terms that are inside.

This visualized example shows WHY the distributive property works for expressions with variables.

\(3(4x+2y-5)=12x+6y-15\)

On the left side, the 3 multiplied by the parentheses means that you have 3 groups and each group has 4 x's, 2 y's, and 5 negative 1's inside it.

On the right side, the expression has been simplified because all the groups have been combined together and you have a total of 12 x's, 6 y's, and 15 negative 1's. 

Left Side
\(3(4x+2y-5)\)

Right Side
\(12x+6y-15\)

These expressions are equivalent because they represent the same amount. They are just written in different forms.

How to Use the Distributive Property

The distributive property can be used to simplify any expression that has something multiplied by a group of terms inside parentheses.

To use the distributive property...

  1. Identify the terms inside the parentheses.
  2. Multiply each of the terms individually.
  3. Add all of the terms together with the correct signs.

When I am identifying the terms in Step 1, I like to think of any subtraction as adding a negative. This makes it easier to identify the right sign when you have to multiply positive and negative numbers.

Example: Simplify \(3x^{2}(4x-2)\)

Step 1: Identify the terms inside the parentheses.

The terms inside of the parentheses are 4x and -2

Step 2: Multiply each of the terms individually.

\(3x^{2} (4x) = 12x^{3}\)

\(3x^{2} (-2) = -6x^{2}\)

Step 3: Add all of the terms together with the correct signs.

\((12x^{3}) + (- 6x^{2})=12x^{3} - 6x^{2}\)

Answer: \(3x^{2}(4x-2)=12x^{3} - 6x^{2}\)

Distributive Property of Division

The distributive property of division is a special case of the distributive property of multiplication. 

\(\frac{a+b+c+d+...}{x}=\frac{a}{x}+\frac{b}{x}+\frac{c}{x}+\frac{d}{x}+...\)

It basically says that if you have a group of terms divided by another term, then you can simplify the expression by dividing each term individually. 

It works because dividing a+b+c+d+... by x is the same as multiplying a+b+c+d+... by \(\frac{1}{x}\), which can be simplified with the distributive property of multiplication.

Even though there aren't any written parentheses, the division bar is a grouping symbol so it acts like a set of parentheses.

Example: Simplify \(\frac{8x^{2}-12x+6}{2x}\)

Step 1: Identify the terms above the division bar.

The terms above the division bar are \(8x^{2}\), -12x, and 6.

Step 2: Divide each of the terms individually.

\(\frac{8x^{2}}{2x} = 4x\)

\(\frac{-12x}{2x} = -6\)

\(\frac{6}{2x} = \frac{3}{x}\)

Step 3: Add all of the terms together with the correct signs.

\((4x)+(-6)+(\frac{3}{x})=4x-6+\frac{3}{x}\)

Answer: \(\frac{8x^{2}-12x+6}{2x}=4x-6+\frac{3}{x}\)

NOTE: This property does NOT apply if you have an expression like this...

\(\frac{x}{a+b+c+d+...}\) 

Reversed Distributive Property

The distributive property of multiplication can also be used in reverse.

\(xa+xb+xc+xd+...=x(a+b+c+d+...)\)

This is especially useful when you are factoring polynomials or converting expressions in expanded form to factored form. 

To use the reversed distributive property...

  1. Identify all the terms in the expression.
  2. Find a common factor that all the terms can be divided by.
  3. Divide each term by that common factor.
  4. Write the common factor outside parentheses and the divided terms inside parentheses.

If you want to fully factor an expression, make sure that the common factor you choose is the greatest common factor. 

Example: Write \(6x^{2}+15x\) in factored form.

Step 1: Identify all the terms in the expression.

The terms in the expression are \(6x^{2}\) and 15x.

Step 2: Find a common factor that all the terms can be divided by.

Both terms can be divided by 3x.

Step 3: Divide each term by the common factor. 

\(\frac{6x^{2}}{3x}=2x\)

\(\frac{15x}{3x}=5\)

Step 4: Write the common factor outside parentheses and the divided terms inside parentheses.

\(3x(2x+5)\)

Answer: \(6x^{2}+15x=3x(2x+5)\)

When you use the reversed distributive property, you can use the distributive property to check your answers.

In this example, we could say, "3x times 2x gives us \(6x^{2}\)...and 3x times 5 gives us \(15x\)...so, that means we used the reversed distributive property correctly."

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