While it was partially covered in previous lessons and used in the last lesson, we have never fully gone in-depth and seen the use of the distributive property. Say you have the expression 4(2+3). In this, you would distribute the 4 and therefore multiply both terms in the expression by the outside term, which in this case would be 4. Therefore, when distributing, it would look like 4(2)+4(3), or 8+12, and therefore 20. Another thing to think about when distributing is when variables come into play. When distributing a variable, such as x(x+5), the x will multiply by another x, which is the same as squaring, which would therefore give x², and multiplying x by 5 would simply give you 5x. Additionally, if multiplying by a variable and a number like 5x, you will then apply both multiplying the terms by 5 and multiplying the terms by x. Also, if multiplying a variable and a term and a different variable (or vice versa), like 8x(2y), you would just put 16xy due to the x and y not being able to combine in any other way. Finally, if distributing a variable with an exponent as well, like x²(4x), that would give you 4x³, because you're just multiplying x² by x, aka x·x·x, which would be x³. Another way to think about it would be adding the exponents together, due to the 2 on x² and assumed 1 on x, which would, of course, add to 3. Now that you have an overview of the distributive property, let's get into some examples where you will see these different cases I mentioned.
1: Simplify 2(5x+2)+8
Firstly, we can visually distribute the 2, giving us 2(5x)+2(2)+8. Next, we can multiply 2 and 5x as well as 2 and 2 to give us the expression 10x+4+8. Finally, we can add 4 and 8 due to them being like terms, giving us our final answer of 10x+12.
Answer: 10x+12
2: Simplify x(3+5x)
In this expression, we are adding on another layer to distributing, with us now needing to distribute a variable. When visually applying/distributing the x, we get x(3)+x(5x). For the first multiplication, x·3 simply equals 3x as you've seen before. However, with multiplying 5x by x, you get 5x². This can be logically thought of because you are basically multiplying the x in 5x by itself, which is the same meaning as squaring something. Therefore, our final answer is 3x+5x².
Answer: 3x+5x²
3: Simplify 4x(2x+9)
Yet again, we are seeing another level being added onto distributing. With this equation, you would multiply the terms in the parentheses by both 4 and x. With multiplying 4x and 2x, you would get 8x², due to the 4 and 2 multiplying to get 8 and the x's multiplying together which would result in x². For the second term, 4x·9 would simply equal 36x, due to the 9 and 4 multiplying together and the x staying present but not affecting anything. Therefore, the simplified version of the expression is 8x²+36x.
Answer: 8x²+36x
4: Simplify 2.5x(8y+4x-2)
In this expression, we now have 2 variables, those being x and y. When multiplying 2.5x and 8y, you simply get 20xy, with the 2.5 and 8 multiplying to get the product of 20, and the x just being placed by the y, due to them not being able to be combined in anyway. Multiplying 2.5x and 4x was another form of what you saw in example 3, and the 2.5 and 4 give a product of 10 and the x and x giving x², meaning we got 10x² for the second distribution. Finally, 2.5x and -2 are multiplied to get -5x, with our final answer being 20xy+10x²-5x.
Answer: 20xy+10x²-5x.
5: Simplify -7x²(4.5x+9y-x³)
In this expression, we have a negative number as the coefficient as well as an exponent on the variable! When distributing it visually, it's broken down to -7x²(4.5x)+-7x²(9y)+-7x²(-x³). When looking at -7x²(4.5x), we can first multiply the -7 and 4.5 to get -31.5, and then multiply the x² and x to get x³, giving us -31.5x³ overall. When looking at the next distribution of -7x²(9y), we can follow similar procedures with multiplying the -7 and 9 to get the product of -63, then multiply x² and y to simply get x²y. Putting them together, we have -63x²y. Next, we have -7x²(-x³), which we can multiply the -7 and -1 to get 7, then combine x² and x³ to get x⁵, giving us the third term of 7x⁵. Finally, all the terms can be combined together to get our answer of -31.5x³-63x²y+7x⁵.
Answer: -31.5x³-63x²y+7x⁵
6: Simplify -3.5x²y(2x-4y+8)
When distributing the term, you get -3.5x²y(2x)+-3.5x²y(-4y)+-3.5x²y(8). For the first multiplication of -3.5x²y and (2x), -3.5 and 2 will be multiplied to get -7, x² and x will be multiplied to get x³, and the y will simply carry over, giving us -7x³y. For the second multiplication of -3.5x²y(-4y), the -3.5 and -4 multiply to get positive 14, the y's multiply with each other to get y², and the x² just carries over to get 14y²x². Finally, -3.5x²y and 8 multiply over with the -3.5 and 8 multiplying to get -28 and the x²y carries over. Therefore, our final answer is -3.5x²y+14y²x²-28x²y.
Answer: -3.5x²y+14y²x²-28x²y
1: Simplify 4(3.5x-6)
2: Simplify 7.35(6.25y-3y)+4x
3: Simplify 9x(3x-5y)+3.3x
4: Simplify -2y(7.7+4.5y+33)
5: Simplify 11xy(6y-2x+9.5)
6: Simplify 5.35y³(-4y+31-0.5x)
7: Simplify -14x²(9x-8y+0)
8: Simplify 43xx(-0.46xxy+7.8x)
9: Simplify -9y⁴(3y-7.1x)-44
10: Simplify -6.5xz³(8y-13z+4.4)
11: Simplify --3.2x(9x+0z³-22y)
12: Simplify 2³x³(4y-z)+3y
1: 14x-24
2: 23.89y+4x
3: 27x²-45xy+3.3x
4: -9y²-81.4y
5: 66xy²-22x²y+104.5xy
6: -21.4y⁴+165.85y³-2.68y³x
7: -126x³+112x²y
8: -19.78x⁴y+335.4x³
9: -27y⁵+63.9y⁴x-44
10: -52xyz³+84.5xz⁴
11: 28.8x²-70.4xy
12: 32x³y-8x³z+3y