The Distributive Property

When adding real numbers we have the freedom to associate the numbers involved smoothly, the same happens if we are multiplying real numbers, however, we must be cautious when we come across mixed operations, that is, sums and products at the same time. We will see a property that allows us to operate sums and products at the same time:

The distributive property states that if a number multiplies the sum of two numbers, then the factor involved is distributed among each of the addends. Formally, if a, b and c are real numbers, then

We can also apply this property if a subtraction is involved instead of an addition within the parentheses, as follows:

We notice that if we observe this equality from right to left, we are taking the common factor that exists in both addends and we are taking it out to multiply:

a \cdot b \pm a \cdot c = a \cdot (b \pm c)

This is one of the most used properties in the calculation of mixed operations and from them, some cases are deduced that facilitate the simplification of mathematical expressions. Let’s see some examples to understand this property well:

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Examples

Example 1

Use the distributive property to expand the expression 2 \cdot (1 + 6). In this case, it is not necessary to use the distributive property since we can add the numbers that are inside the parentheses and then multiply in the following way:

2 \cdot (1 + 6) = 2 \cdot 7 = 14

Example 2

Use the distributive property to expand the expression 2 \cdot \left (1 + \sqrt {6} \right). Note that one of the addends involved is the square root of 6, therefore it cannot be added with 1, so we distribute the factor involved

2 \cdot \left( 1 + \sqrt{6} \right) = 2 \cdot 1 + 2 \cdot \sqrt{6} = 2 + 2 \sqrt{6}

Example 3

Use the distributive property to expand the expression 5 \cdot \left (x - \sqrt {10} \right) . Note that one of the addends involved is the square root of 10 and the other is an unknown, therefore they cannot be subtracted, so we distribute the factor involved

5 \cdot \left( x - \sqrt{10} \right) = 5 \cdot x - 5 \cdot \sqrt{10} = 5x - 5\sqrt{10}

Example 4

Use the distributive property to expand the expression x \cdot \left (x + x^2 \right) . Note that one of the addends involved is an unknown and the other is an unknown squared, therefore they cannot be added, then we distribute the factor involved

x \cdot \left( x + x^2 \right) = x \cdot x + x \cdot x^2 = x^2 + x^3

Example 5

Use the distributive property to take out the common factor of the expression 18 + 3 \sqrt {7}. Note that 18 = 3 \cdot 6, then,

18 + 3\sqrt{7} = 3 \cdot 6 + 3 \sqrt{7} = 3 \cdot \left( 6 + \sqrt{7} \right)

Example 6

Use the distributive property to take out the common factor of the expression x^4 - 8x. Note that one of the addends involved is an unknown raised to four and the other is 8 times said unknown, therefore they cannot be subtracted, then

x^4 - 8x = x \cdot x^3 - x \cdot 8 = x \cdot \left( x^3 - 8 \right)

Example 7

Use the distributive property to take the common factor of the expression 12x^7 + 15x^4. These two elements cannot be added, so

12x^7 + 15x^4 = 3 \cdot 4 \cdot x^4 \cdot x^3 + 3 \cdot 5 \cdot x^4 = 3 x^4 \cdot \left( 4x^3 + 5 \right)


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