Extended Distributive Property - Examples, Exercises and Solutions

The extended distributive property allows us to solve exercises with two sets of parentheses that are multiplied by eachother.

For example: (a+1)×(b+2) (a+1)\times(b+2)

To find the solution, we will go through the following steps:

  • 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.

ab+2a+b+2 ab+2a+b+2

The Extended Distributive Property

Suggested Topics to Practice in Advance

  1. Algebraic Method
  2. Factorization: Common factor extraction

Practice Extended Distributive Property

Exercise #1

(3+20)×(12+4)= (3+20)\times(12+4)=

Video Solution

Step-by-Step Solution

Simplify this expression paying attention to the order of arithmetic operations which states that exponentiation precedes multiplication and division before addition and subtraction and that parentheses precede all of them.

Therefore, let's first start by simplifying the expressions within parentheses, then we perform the multiplication between them:

(3+20)(12+4)=2316=368 (3+20)\cdot(12+4)=\\ 23\cdot16=\\ 368 Therefore, the correct answer is option A.

Answer

368

Exercise #2

(12+2)×(3+5)= (12+2)\times(3+5)=

Video Solution

Step-by-Step Solution

Simplify this expression paying attention to the order of arithmetic operations which states that exponentiation precedes multiplication and division before addition and subtraction and that parentheses precede all of them.

Therefore, let's start by simplifying the expressions within parentheses, then perform the multiplication between them:

(12+2)(3+5)=148=112 (12+2)\cdot(3+5)= \\ 14\cdot8=\\ 112 Therefore, the correct answer is option C.

Answer

112

Exercise #3

(35+4)×(10+5)= (35+4)\times(10+5)=

Video Solution

Step-by-Step Solution

We will open the parentheses using the extended distributive property and create a long addition exercise:

We multiply the first term of the left parenthesis by the first term of the right parenthesis.

Then we multiply the first term of the left parenthesis by the second term of the right parenthesis.

Now we multiply the second term of the left parenthesis by the first term of the left parenthesis.

Finally, we multiply the second term of the left parenthesis by the second term of the right parenthesis.

In the following way:

(35×10)+(35×5)+(4×10)+(4×5)= (35\times10)+(35\times5)+(4\times10)+(4\times5)=

We solve each of the exercises within parentheses:

350+175+40+20= 350+175+40+20=

We solve the exercise from left to right:

350+175=525 350+175=525

525+40=565 525+40=565

565+20=585 565+20=585

Answer

585

Exercise #4

(a+4)(c+3)= (a+4)(c+3)=

Video Solution

Step-by-Step Solution

When we encounter a multiplication exercise of this type, we know that we must use the distributive property.

Step 1: Multiply the first factor of the first parentheses by each of the factors of the second parentheses.

Step 2: Multiply the second factor of the first parentheses by each of the factors of the second parentheses.

Step 3: Group like terms.

 

a * (c+3) =

a*c + a*3

4  * (c+3) =

4*c + 4*3

 

ac+3a+4c+12

 

There are no like terms to simplify here, so this is the solution!

Answer

ac+3a+4c+12 ac+3a+4c+12

Exercise #5

It is possible to use the distributive property to simplify the expression?

If so, what is its simplest form?

(a+c)(4+c) (a+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+yd (\textcolor{red}{x}+\textcolor{blue}{y})(t+d)=\textcolor{red}{x}t+\textcolor{red}{x}d+\textcolor{blue}{y}t+\textcolor{blue}{y}d Keep 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)x4+xc+c4+cc4x+xc+4c+c2 (\textcolor{red}{x}+\textcolor{blue}{c})(4+c)\\ \textcolor{red}{x}\cdot 4+\textcolor{red}{x}\cdot c+\textcolor{blue}{c}\cdot 4+\textcolor{blue}{c} \cdot c\\ 4x+xc+4c+c^2 To simplify this expression, we use the power law for multiplication between terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+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+4c \textcolor{purple}{4x}\textcolor{green}{+xc}\textcolor{black}{+4c}\textcolor{orange}{+c^2 }\\ \textcolor{orange}{c^2 }\textcolor{green}{+xc}\textcolor{purple}{+4x}\textcolor{black}{+4c}\\ We 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 4x+cx+4c+c^2

Exercise #1

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

(ab)(cd) (ab)(c d)

Video Solution

Step-by-Step Solution

Let's remember the extended distributive property:

(a+b)(c+d)=ac+ad+bc+bd (\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d Note that the operation between the terms inside the parentheses is a multiplication operation:

(ab)(cd) (a b)(c d) 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)=abcd (a b)(c d)= \\ abcd Therefore, 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 abcd .

Exercise #2

(2xy)(43x)= (2x-y)(4-3x)=

Video Solution

Step-by-Step Solution

Let's simplify the given expression, develop parentheses using the expanded distributive law:

(a+b)(c+d)=ac+ad+bc+bd (\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d Let's pay attention that in the expression on the right side of the distributive law we take as a factor the first term that the parentheses are multiplied by, so we will not forget that the sign before the term is an inseparable part of it, and we will also apply the laws of sign multiplication and thus we can present any expression in parentheses, which we will develop using the above expression, starting as an expression in which the multiplication operation takes place between all the terms:

(2xy)(43x)(2x+(y))(4+(3x)) (2x-y)(4-3x)\\ (\textcolor{red}{2x}+\textcolor{blue}{(-y)})(4+(-3x))\\ Let's start then by opening the parentheses:

(2x+(y))(4+(3x))2x4+2x(3x)+(y)4+(y)(3x)8x6x24y+3xy (\textcolor{red}{2x}+\textcolor{blue}{(-y)})(4+(-3x))\\ \textcolor{red}{2x}\cdot 4+\textcolor{red}{2x}\cdot(-3x)+\textcolor{blue}{(-y)}\cdot 4+\textcolor{blue}{(-y)} \cdot(-3x)\\ 8x-6x^2-4y+3xy In the calculation of the multiplications above we used the multiplication table and the sign multiplication laws for multiplication between terms with identical bases:

aman=am+n a^m\cdot a^n=a^{m+n}

In the next step we will enter similar terms, we will define similar terms as terms in which the unknown (or unknowns, each separately), in this case, x and y, have identical powers (in the absence of one of the unknowns from the expression, we will relate to its power as zero power, since raising any number to the power of zero will yield the result 1), we will use the addition law in multiplication, in addition we will arrange the expression from the highest power to the lowest from left to right (we will relate to the free term as the power of zero),

Let's pay attention that in the expression we received in the last step there are four different terms, since there is not even one pair of terms in which the unknowns (the variables) have the same power, so the expression we already received, is the final and most simplified expression, so we will settle for arranging it again from the highest power to the lowest from left to right:
8x6x24y+3xy6x2+3xy+8x4y \textcolor{purple}{ 8x}\textcolor{green}{-6x^2}-4y\textcolor{orange}{+3xy}\\ \textcolor{green}{-6x^2}\textcolor{orange}{+3xy}\textcolor{purple}{ +8x}-4y\\ We highlighted the different terms using colors, and as already emphasized before, we made sure that the sign before the term is an inseparable part of it,

We thus received that the correct answer is answer D.

Answer

6x2+3xy+8x4y -6x^2+3xy +8x-4y

Exercise #3

(2x3)×(5x7) (2x-3)\times(5x-7)

Video Solution

Step-by-Step Solution

To answer this exercise, we need to understand how the extended distributive property works:

For example:

(a+1)∗(b+2)

To solve this type of exercises, the following steps must be taken:

Step 1: multiply the first factor of the first parentheses by each of the factors of the second parentheses.

Step 2: multiply the second factor of the first parentheses by each of the factors of the second parentheses.

Step 3: group like terms together.

 

ab∗2ab∗2

 

We start from the first number of the exercise: 2x

2x*5x+2x*-7

10x²-14x

 

We will continue with the second factor: -3

-3*5x+-3*-7

-15x+21

 

We add all the data together:

 

10x²-14x-15x+21

10x²-29x+21

 

Answer

10x229x+21 10x^2-29x+21

Exercise #4

It is possible to use the distributive property to simplify the expression

(17+c)(5+a+3) (17+c)(5+a+3)

Video Solution

Step-by-Step Solution

We can use the right parenthesis since it can be simplified in the following way:

(8+a)

Then we will get the exercise:

(17+c)(8+a)= (17+c)(8+a)=

136+17a+8c+ca 136+17a+8c+ca

Answer

Yes, 136+17a+8c+ca 136+17a+8c+ca

Exercise #5

Is it possible to use the distributive property to simplify the expression?

If so,what is its simplest form?

(3a4)b+2 (3a-4)b+2

Video Solution

Step-by-Step Solution

We simplify the expression by opening the parentheses using the distributive property:

x(y+z)=xy+xz x(y+z)=xy+xz Note that in the distributive property formula we assume that there is addition between the terms inside the parentheses, therefore, of course, we will not forget that the sign of the coefficient of the term is very importantt.

Also, we apply the rules of multiplication of signs and so we can present any expression within parentheses, which is opened with the help of the distributive property, as an expression in which there is an addition operation between all the terms:

(3a4)b+2(3a+(4))b+2 (3a-4)b+2\\ \big(3a+(-4)\big)b+2 We continue and open the parentheses using the distributive property:

(3a+(4))b+23ab+(4)b+23ab4b+2 \big(3a+(-4)\big)b+2\\ 3a\cdot b+(-4)\cdot b +2\\ 3ab-4b+2 Therefore, the correct answer is option c.

Answer

No, 3ab4b+2 3ab-4b+2

Exercise #1

(7x+3)×(10+4)=238 (7x+3)\times(10+4)=238

Video Solution

Step-by-Step Solution

First, we solve the addition exercise in the right parenthesis:

(7x+3)+14=238 (7x+3)+14=238

Now, we multiply each of the terms inside the parenthesis by 14:

(14×7x)+(14×3)=238 (14\times7x)+(14\times3)=238

We solve each of the exercises inside the parenthesis:

98x+42=238 98x+42=238

We move the sections and keep the appropriate sign:

98x=23842 98x=238-42

98x=196 98x=196

We divide the two parts by 98:

9898x=19698 \frac{98}{98}x=\frac{196}{98}

x=2 x=2

Answer

2

Exercise #2

(9+17x)×(6+1)=420 (9+17x)\times(6+1)=420

Calculate a X

Video Solution

Step-by-Step Solution

First, we solve the addition exercise in the right parenthesis:

(9+17x)×7=420 (9+17x)\times7=420

Now, we multiply each of the terms inside the parenthesis by 7:

(9×7)+(17x×7)=420 (9\times7)+(17x\times7)=420

We solve each of the exercises inside the parenthesis:

63+119x=420 63+119x=420

We move the sections and keep the appropriate sign:

119x=42063 119x=420-63

119x=357 119x=357

We divide the two parts by 119:

119119x=357119 \frac{119}{119}x=\frac{357}{119}

x=3 x=3

Answer

3

Exercise #3

(a+3a)×(5+2)=112 (a+3a)\times(5+2)=112

Calculate a a

Video Solution

Step-by-Step Solution

First, we solve the two exercises in parentheses:

4a×7=112 4a\times7=112

Divide each of the sections by 4:

4a×74=1124 \frac{4a\times7}{4}=\frac{112}{4}

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

a×7=28 a\times7=28

Remember that:

a×7=a7 a\times7=a7

Divide both sections by 7:

a77=287 \frac{a7}{7}=\frac{28}{7}

a=4 a=4

Answer

4

Exercise #4

Is the equation correct?

a2+9a20=(a+4)(a5) a^2+9a-20=(a+4)(a-5)

Video Solution

Step-by-Step Solution

We solve the right side of the equation using the extended distributive property:(a+b)×(c+d)=ac+ad+bc+bd (a+b)\times(c+d)=ac+ad+bc+bd

(a+4)(a5)=a25a+4a20 (a+4)(a-5)=a^2-5a+4a-20

a2a20 a^2-a-20

That is, answer D is the correct one.

Answer

No, a -a instead of +9a +9a

Exercise #5

Match the expressions (numbers) with the equivalent expressions (letters):

  1. (2xy)(x+3) (2x-y)(x+3)

  2. (y2x)(3x) (y-2x)(3-x)

  3. (2x+y)(x3) (2x+y)(x-3)

    a.2x26x+yx3y 2x^2-6x+yx-3y

    b.2x26xyx+3y 2x^2-6x-yx+3y

    c.2x2+6xyx3y 2x^2+6x-yx-3y

Video Solution

Step-by-Step Solution

Simplify the given expressions, open parentheses using the extended distributive property:

(a+b)(c+d)=ac+ad+bc+bd (\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d Keep in mind that in the formula form for the distributive property mentioned above, we assume by default that the operation between the terms inside the parentheses is an addition, therefore, of course, we will not forget that the sign of the term's coefficient is an inseparable part of it. Furthermore, we will apply the rules of sign multiplication and thus we can present any expression within parentheses, which is opened with the help of the previous formula, first, as an expression in which an addition operation takes place among all the terms (if necessary),

Then we will simplify each and every one of the expressions of the given problem, respecting the above, first opening the parentheses through the previously mentioned distributive property. Then we will use the substitution property in addition and multiplication before introducing like terms (if there are like terms in the expression obtained after opening the parentheses):

  1. (2xy)(x+3)(2x+(y))(x+3)2xx+2x3+(y)x+(y)32x2+6xyx3y (2x-y)(x+3) \\ \downarrow\\ \big(2x+(-y)\big)(x+3) \\ 2x\cdot x+2x\cdot 3+(-y)\cdot x+(-y)\cdot3\\ \boxed{2x^2+6x-yx-3y}\\

  2. (y2x)(3x)(y+(2x))(3+(x))y3+y(x)+(2x)3+(2x)(x)3yxy6x+2x2 (y-2x)(3-x) \\ \downarrow\\ \big(y+(-2x)\big)\big(3+(-x)\big) \\ y\cdot 3+y\cdot (-x)+(-2x)\cdot 3+(-2x)\cdot(-x)\\ \boxed{3y-xy-6x+2x^2}\\

  3. (2x+y)(x3)(2x+y)(x+(3))2xx+2x(3)+yx+y(3)2x26x+yx3y (2x+y)(x-3) \\ \downarrow\\ (2x+y)(x+(-3)) \\ 2x\cdot x+2x\cdot (-3)+y\cdot x+y\cdot(-3)\\ \boxed{2x^2-6x+yx-3y}\\ As you can notice, in all the expressions where we applied multiplication between the expressions in the previous parentheses, the result of the multiplication (obtained after applying the previously mentioned distributive property) produced an expression in which terms cannot be added, and this is because all the terms in the resulting expression are different from each other (remember that all like variables must be identical and in the same power),

    Now, let's use the substitution property in addition and multiplication to distinguish that:

    The simplified expression in 1 corresponds to the expression in option C,

    The simplified expression in 2 corresponds to the expression in option B,

    The simplified expression in 3 corresponds to the expression in option A,

Therefore, the correct answer (among the options offered) is option B.

Answer

1-b, 2-c, 3-a