It is possible to use the distributive property to simplify the expression below?
What is its simplified form?
\( (ab)(c d) \)
\( \)
Incorrect
Correct Answer:
No, \( abcd \).
Practice more now
Exercises to practice the distributive property
(x−4)×(x−2)=x2−2x−4x+8=x2−6x+8
(x+3)×(x+6)=x2+6x+3x+18=x2+9x+18
The distributive property allows us to remove parentheses and simplify an expression, even if there is more than one set of parentheses.
In order to get rid of of the parentheses, we will multiply each term of the first parentheses by each term of the second parentheses, paying special attention to the addition/ subtraction signs.
For example:
(5+8)×(7+2)
Using the distributive property, we can simplify the expression.
All we need to do is to multiply each of the terms in the first parentheses by each of the terms in the second parentheses:
(5+8)×(7+2)=
5×7+5×2+8×7+8×2=
35+10+56+16=
117
Basic distributive property
Let's take a moment to remember our basic distributive property.
Below we can see the formula:
a×(b+c)=ab+ac
Here, we have multiplied a by each of the terms inside the parentheses, keeping the same order.
Extended distributive property
Now we will apply the same concept in the extended distributive property. This allows us to solve exercises with two sets of parentheses.
For example: (a+b)×(c+d)=ac+ad+bc+bd
How does the extended distributive property work?
Step 1: Multiply the first term in the first parentheses by each of the terms in the second parentheses.
Step 2: Multiply the second term in the first parentheses by each of the terms in the second parentheses.
Step 3: Associate like terms.
Example 1
Step 1: Multiply A by each of the terms included in the second parentheses.
Step 2: Multiply 2 by each of the terms included in the second parentheses.
Step 3: Order the terms and combine like terms, if any:
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Test your knowledge
Question 1
It is possible to use the distributive property to simplify the expression?
If so, what is its simplest form?
\( (x+c)(4+c) =\text{?} \)
Incorrect
Correct Answer:
Yes, the meaning is \( 4x+cx+4c+c^2 \)
Question 2
It is possible to use the distributive property to simplify the expression
\( (a+b)(c\cdot g) \)
Incorrect
Correct Answer:
No, \( acg+\text{bcg} \)
Question 3
It is possible to use the distributive property to simplify the expression
\( a(b+c) \)
Incorrect
Correct Answer:
Yes, the answer \( ab+ac \)
Example 2: What do we do with a minus sign?
So, what do we do when we see a minus sign in one or both of the parentheses? Do we do anything different?
The method is the same! The only difference is that we need to make sure to put out minus/ negative signs in the right places when we distribute.
It can helpful to remember that a "minus sign" is the same as saying "plus a negative number."
For example, 4−2=4+(−2)=2
Let's look at the exercise:
Step 1: Multiply A by each of the terms included inside the second parentheses.
Step 2: Multiply 5 by each of the terms included inside the second parentheses.
Pay attention to the signs of each of the terms! For example, we will see that, −5 times −3 equals +15.
In this case, there are no terms that we want to combine.
Example 3
Task:
Find the value of X:
(X+2)2=(X+5)×(X−2)
Let's look at the left side of the equation and simplify:
(X+2)2=(X+2)×(X+2)
Now we can use the extended distributive property on each side of the equation.
Now the equation looks like this:
(X+2)×(X+2)=(X+5)×(X−2)
After applying the distributive property:
X2+2X+2X+4=X2–2X+5X–10
Let's reduce, combine like terms and arrange the equation.
We will get:
X=−14
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Do you know what the answer is?
Question 1
Expand the following expression:
\( (x+4)(x+3)= \)
Incorrect
Correct Answer:
\( x^2+7x+12 \)
Question 2
Solve the following problem:
\( (a+15)(5+a)= \)
Incorrect
Correct Answer:
\( a^2+20a+75 \)
Question 3
Solve the following problem:
\( (12-x)(x-3)= \)
Incorrect
Correct Answer:
\( 15x-36-x^2 \)
Exercises using the distributive property
Exercise 1
Assignment:
A painter has a canvas with the following dimensions:
(23x+12) length
(20x+7) width
What is the area the painter needs to paint?
Solution:
We multiply the length of the canvas by the width to find the area.
(23x+12)×(20x+7)=
Multiply each term in the first parentheses by each term in the second parentheses.
23x×20x+23x×7+12×20x+12×7=
We solve accordingly
460x2+161x+240x+84=
460x2+401x+84
Answer:
460x2+401x+84
Exercise 2
Task:
Find the area of the following rectangle:
Leave variables in your answer.
Solution:
To find the area we multiply the width by the length.
3y×(y+3z)=
Multiply 3y by each of the terms in parentheses.
3y×y+3y×3z=
Solve accordingly
3y2+9yz
Answer:
3y2+9yz
Check your understanding
Question 1
Solve the following problem:
\( (x+2)(x-4)= \)
Incorrect
Correct Answer:
\( x^2-2x-8 \)
Question 2
Solve the following problem:
\( (x-6)(x+8)= \)
Incorrect
Correct Answer:
\( x^2+2x-48 \)
Question 3
Solve the following problem:
\( (x-8)(x+y)= \)
Incorrect
Correct Answer:
\( x^2+xy-8x-8y \)
Exercise 3
Task:
(3+20)×(12+4)=
Solution:
We multiply each of the terms in the first parentheses by the terms in the second parentheses.
3×12+3×4+20×12+20×4=
Solve accordingly
36+12+240+80=
We add everything together
48+320=368
Answer:
368
Exercise 4
Task:
(12+2)×(3+5)=
Solution:
We multiply each of the terms in the first parentheses by the terms of the second parentheses.
12×3+12×5+2×3+2×5=
Solve accordingly
36+60+6+10=
We add everything together
96+16=112
Answer:
112
Do you think you will be able to solve it?
Question 1
Resolve -
\( (x-3)(x-6)= \)
Incorrect
Correct Answer:
\( x^2-9x+18 \)
Question 2
Solve the exercise:
\( (2y-3)(y-4)= \)
Incorrect
Correct Answer:
\( 2y^2-11y+12 \)
Question 3
Solve the exercise:
\( (3x-1)(x+2)= \)
Incorrect
Correct Answer:
\( 3x^2+5x-2 \)
Exercise 5
Task:
(7x+4)×(3x+4)=
Solution:
We multiply each of the terms of the first parentheses by the terms of the second parentheses.
7x×3x+7x×4+4×3x+4×4=
Solve accordingly
21x2+28x+12x+16=
21x2+40x+16
Answer:
21x2+40x+16
Exercise 6
Task:
(2x−3)×(5x−7)
We multiply each of the terms of the first parentheses by the terms of the second parentheses.
2x×5x+2x×(−7)+(−3)×5x+(−3)×(−7)=
Solve accordingly
10x2−14x−15x+21=
10x2−29x+21
Answer:
10x2−29x+21
Test your knowledge
Question 1
Solve the exercise:
\( (5x-2)(3+x)= \)
Incorrect
Correct Answer:
\( 5x^2+13x-6 \)
Question 2
\( (12+2)\times(3+5)= \)
Incorrect
Correct Answer:
112
Question 3
It is possible to use the distributive property to simplify the expression below?
What is its simplified form?
\( (ab)(c d) \)
\( \)
Incorrect
Correct Answer:
No, \( abcd \).
Review questions
What is the distributive property of multiplication?
The distributive property of multiplication is a rule in mathematics that says that multiplying the sum of two (or more) numbers is the same as multiplying the numbers separately and adding/ subtracting them together.
Distributive property of multiplication over addition:
a×(b+c)=a×b+a×c
Distributive property of multiplication over subtraction:
a×(b−c)=a×b−a×c
What is the distributive property of division?
Just as in the distributive property of multiplication, the distributive property of division (also over addition or subtraction) helps us to simplify an expression.
We can express it as follows:
(a+b):c=a:c+b:c
Do you know what the answer is?
Question 1
It is possible to use the distributive property to simplify the expression?
If so, what is its simplest form?
\( (x+c)(4+c) =\text{?} \)
Incorrect
Correct Answer:
Yes, the meaning is \( 4x+cx+4c+c^2 \)
Question 2
It is possible to use the distributive property to simplify the expression
\( (a+b)(c\cdot g) \)
Incorrect
Correct Answer:
No, \( acg+\text{bcg} \)
Question 3
It is possible to use the distributive property to simplify the expression
\( a(b+c) \)
Incorrect
Correct Answer:
Yes, the answer \( ab+ac \)
What is the extended distributive property?
The extended distributive property uses the same concept as the basic distributive property to simplify expressions with two sets of parentheses.
Where is the extended distributive property used?
Example 1
Task:
Solve (x+3)(x−8)=
We will use the extended distributive property, multiplying each of the terms as follows:
(x+3)(x−8)=x2−8x+3x−24
Reducing like terms we get
(x+3)(x−8)=x2−5x−24
Answer
x2−5x−24
Example 2
Task:
(2x−1)(3x−5)=
Using the extended distributive property we get:
(2x−1)(3x−5)=6x2−10x−3x+5
Reducing like terms:
(2x−1)(3x−5)=6x2−13x+5
Answer
6x2−13x+5
Check your understanding
Question 1
Expand the following expression:
\( (x+4)(x+3)= \)
Incorrect
Correct Answer:
\( x^2+7x+12 \)
Question 2
Solve the following problem:
\( (a+15)(5+a)= \)
Incorrect
Correct Answer:
\( a^2+20a+75 \)
Question 3
Solve the following problem:
\( (12-x)(x-3)= \)
Incorrect
Correct Answer:
\( 15x-36-x^2 \)
Examples with solutions for Extended Distributive Property
Exercise #1
It is possible to use the distributive property to simplify the expression below?
What is its simplified form?
(ab)(cd)
Video Solution
Step-by-Step Solution
Let's remember the extended distributive property:
(a+b)(c+d)=ac+ad+bc+bdNote that the operation between the terms inside the parentheses is a multiplication operation:
(ab)(cd)Unlike in the extended distributive property previously mentioned, which is addition (or subtraction, which is actually the addition of the term with a minus sign),
Also, we notice that since there is a multiplication among all the terms, both inside the parentheses and between the parentheses, this is a simple multiplication and the parentheses are actually not necessary and can be remoed. We get:
(ab)(cd)=abcdTherefore, opening the parentheses in the given expression using the extended distributive property is incorrect and produces an incorrect result.
Therefore, the correct answer is option d.
Answer
No, abcd.
Exercise #2
It is possible to use the distributive property to simplify the expression?
If so, what is its simplest form?
(x+c)(4+c)=?
Video Solution
Step-by-Step Solution
We simplify the given expression by opening the parentheses using the extended distributive property:
(x+y)(t+d)=xt+xd+yt+ydKeep in mind that in the distributive property formula mentioned above, we assume that the operation between the terms inside the parentheses is an addition operation, therefore, of course, we will not forget that the sign of the term's coefficient is ery important.
We will also apply the rules of multiplication of signs, so we can present any expression within parentheses that's opened with the distributive property as an expression with addition between all the terms.
In this expression we only have addition signs in parentheses, therefore we go directly to opening the parentheses,
We start by opening the parentheses:
(x+c)(4+c)x⋅4+x⋅c+c⋅4+c⋅c4x+xc+4c+c2To simplify this expression, we use the power law for multiplication between terms with identical bases:
am⋅an=am+n
In the next step like terms come into play.
We define like terms as terms in which the variables (in this case, x and c) have identical powers (in the absence of one of the variables from the expression, we will refer to its power as zero power, this is because raising any number to the power of zero results in 1).
We will also use the substitution property, and we will order the expression from the highest to the lowest power from left to right (we will refer to the regular integer as the power of zero),
Keep in mind that in this new expression there are four different terms, this is because there is not even one pair of terms in which the variables (different) have the same power. Also it is already ordered by power, therefore the expression we have is the final and most simplified expression:4x+xc+4c+c2c2+xc+4x+4cWe highlight the different terms using colors and, as emphasized before, we make sure that the main sign of the term is correct.
We use the substitution property for multiplication to note that the correct answer is option A.
Answer
Yes, the meaning is 4x+cx+4c+c2
Exercise #3
It is possible to use the distributive property to simplify the expression
(a+b)(c⋅g)
Video Solution
Step-by-Step Solution
To solve this problem, we must determine if we can apply the distributive property to simplify the expression (a+b)(c⋅g).
The distributive property states that for any three terms, the expression x(y+z) results in xy+xz. Here, we have the sum (a+b) and the product (c⋅g).
We can treat (c⋅g) as a single term because it involves multiplication, which makes it like a single number or variable in terms of manipulating the expression algebraically. Therefore, using the distributive property, we distribute (c⋅g) over the terms within the parentheses:
Step 1: Distribute c⋅g to a, yielding acg.
Step 2: Distribute c⋅g to b, yielding bcg.
Hence, the simplified expression is:
acg+bcg.
Therefore, the correct answer, according to the choices provided, is:
No, acg+bcg.
Answer
No, acg+bcg
Exercise #4
It is possible to use the distributive property to simplify the expression
a(b+c)
Video Solution
Step-by-Step Solution
To solve the problem and apply the distributive property correctly, follow these steps:
Identify the expression, which is a(b+c).
Apply the distributive property: multiply a by each term inside the parentheses.
Applying this, we get:
a×b=ab
a×c=ac
Combine these two products:
The simplified expression is: ab+ac.
This matches with answer choice 2: Yes, the answer ab+ac.
Answer
Yes, the answer ab+ac
Exercise #5
Expand the following expression:
(x+4)(x+3)=
Video Solution
Step-by-Step Solution
Let's simplify the given expression by opening the parentheses using the extended distribution law:
(a+b)(c+d)=ac+ad+bc+bd
Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition. Therefore we won't forget of course that the sign preceding the term is an inseparable part of it. We will also apply the rules of sign multiplication and thus we can present any expression in parentheses. We'll open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix. Therefore we'll proceed directly to opening the parentheses,
Let's begin:
(x+4)(x+3)x⋅x+x⋅3+4⋅x+4⋅3x2+3x+4x+12
In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:
am⋅an=am+n
In the next step we'll combine like terms, which we define as terms where the variable (or variables each separately), in this case x, have identical exponents .(In the absence of one of the variables from the expression, we'll consider its exponent as zero power given that raising any number to the power of zero yields 1) We'll apply the commutative property of addition, furthermore we'll arrange (if needed) the expression from highest to lowest power from left to right (we'll treat the free number as having zero power): x2+3x+4x+12x2+7x+12In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,