Distributive Property

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Student: What is the distributive property?

Mentor: Well, the distributive property states that the product of the sum or difference of two numbers is the same as the sum or difference of their products. For example, 2*(5-3) is the same value as 2*5 - 2*3. The distributive property is used if there is a number (or variable) that needs to be multiplied by an addition or subtraction problem in parenthesis.

Student: I don't see how 2(5-3) and 2*5 - 2*3 can be the same.

Mentor: Let's first use the distributive property to solve this problem by finding the difference of their products. That means that we will do what?

Student: The 2 outside of the parenthesis needs to be multiplied by the numbers 5 and 3 in the parenthesis?

Mentor: Yes, so the distributive property tells us to multiply the two by each of the numbers and keep the sign that is between the numbers. Therefore, you would first multiply 2 by 5 and what would your answer be?

Student: 2*5=10.

Mentor: Then you would keep the minus sign and put it after the 10. So that it would look like this: 10 - __. Next, you would multiply the 2 by the 3 and what would your answer be?

Student: 2*3 = 6.

Mentor: Correct, so your answer for 2*5 - 2*3 would be: 10 - 6. Now you can do simple subtraction and what do you get?

Student 10-6 = 4. So the answer is four.

Mentor: Let's see if that is the same answer as if we did the subtraction within the parenthesis and then multiplied it by the number outside of the parenthesis so that we are finding the product of the difference.

2(5-3) => 2(2) = 4.

Student: They are the same! Either way that we solve the problem, whether we find the product of the difference of two numbers in parenthesis or the difference of their products, we get the same answer.

Mentor: Exactly. Now let's try using a variable and applying the distributive property. Here is a problem to solve:

-4 + x(3 - x) = ?

Student: I would first use the distributive property so I multiply "x" by 3. This would give me 3x.

Then, I would write down the minus sign after the "x" and then multiply the "x" by the "x" in the parenthesis and that would be x^2.

Mentor: Great, so what would the difference of their products be?

Student: 3x - x^2. Then I have to make sure to include the -4 at the beginning of the problem so the final solution would be: -4+3x-x^2.

Mentor: Good! Now let's try solving for "x" using the distributive property.

Solve for "x" in this problem: 4= -5(x + (-2))

Student: Well first I will do the distributive property so I would start at the beginning of the parenthesis and multiply (-5) by "x" which would be (-5x).

Next, I would write the addition sign that is in the parenthesis between the two values. Then, I will multiply the (-5) by (-2) which is 10!

Mentor: So what have you solved so far?

Student: 4 = (-5x) + 10

Mentor: Now you can solve for "x".

Student: Right, so I want to get "x" on a side by itself. To do this I will subtract 10 from both sides and the problem will look like this:

4 - 10 = (-5x) + 10 - 10 => 4-10 = (-5x)

Then I will solve 4-10 and that is -6. Now my problem is:

-6 = (-5x)

Now I have to get x by itself so I need to divide both sides by (-5) so I have "x" on one side. The solution then is:

6/5 = x

Mentor: Good work! You used the distributive property to solve for "x". Let's check your work: The original problem was 4= -5(x + (-2)) so I will replace "x" with (6/5) since that is the number that you found it to represent.

Here I will use the distributive property and solve to get: 4 = -6 + (10). Now I will do simple addition: 4 = 4. Good work, you found the correct value for "x"!

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