Extended Distributive Property - Examples, Exercises and Solutions

Let's simplify the given expression, open the parentheses using the distributive property:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$Therefore, the correct answer is option A.

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!

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

$(\textcolor{red}{x}+\textcolor{blue}{y})(t+d)=\textcolor{red}{x}t+\textcolor{red}{x}d+\textcolor{blue}{y}t+\textcolor{blue}{y}d$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:

$(\textcolor{red}{x}+\textcolor{blue}{c})(4+c)\\ \textcolor{red}{x}\cdot 4+\textcolor{red}{x}\cdot c+\textcolor{blue}{c}\cdot 4+\textcolor{blue}{c} \cdot c\\ 4x+xc+4c+c^2$To simplify this expression, we use the power law for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

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:$\textcolor{purple}{4x}\textcolor{green}{+xc}\textcolor{black}{+4c}\textcolor{orange}{+c^2 }\\ \textcolor{orange}{c^2 }\textcolor{green}{+xc}\textcolor{purple}{+4x}\textcolor{black}{+4c}\\$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.

Let's remember the extended distributive property:

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

$(a b)(c d)$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:

$(a b)(c d)= \\ abcd$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.

Simplify this expression paying attention to the order of arithmetic operations. Exponentiation precedes multiplication whilst division precedes addition and subtraction. Parentheses precede all of the above.

Therefore, let's first start by simplifying the expressions within the parentheses. Then we can proceed to perform the multiplication between them:

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

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

Thus, let's begin by simplifying the expressions within the parentheses, and following this, the multiplication between them.

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

We begin by opening the parentheses using the extended distributive property to create a long addition exercise:

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

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:

$(35\times10)+(35\times5)+(4\times10)+(4\times5)=$

We solve each of the exercises within parentheses:

$350+175+40+20=$

We solve the exercise from left to right:

$350+175=525$

$525+40=565$

$565+20=585$

We will use the expanded distributive law to simplify the given expression, let's recall it:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses, using the mentioned distributive law, and then we'll combine like terms if possible. We'll do this while paying attention to the correct multiplication of signs:

$(-7-4y)(5x+6)=\\ -35x-42-20xy-24y$Therefore, the correct answer is answer A.

We will use the extended distribution law to simplify the given expression, let's recall it:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses, using the mentioned distribution law, and then we'll combine like terms if possible. We'll do this while paying attention to the correct multiplication of signs:

$(2x+3)(-5-x)= \\ -10x-2x^2-15-3x=\\ \boxed{-2x^2-13x-15}$Therefore, the correct answer is answer D.

We will use the expanded distributive law to simplify the given expression, let's recall it:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses, using the mentioned distributive law, and then we'll combine like terms if possible. We'll do this while paying attention to the correct multiplication of signs:

$(2x-y)(4-3x)= \\ \boxed{8x-6x^2-4y+3xy}$Therefore, the correct answer is answer C.

We will use the expanded distribution law to simplify the given expression, let's recall it:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses using the distribution law mentioned, and then we'll combine like terms if possible. We'll do this while paying attention to the correct multiplication of signs:

$(3b+7a)(-5a+2b)= \\ -15ab+6b^2-35a^2+14ab=\\ \boxed{6b^2-ab-35a^2}$Therefore, the correct answer is answer B.

Let's simplify the given expression by factoring the parentheses using the expanded distributive law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$Note that that the sign before the term is an inseparable part of it.

We will also apply the laws of sign multiplication and thus we can present any term in parentheses to make things simpler.

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

$(\textcolor{red}{2x}+\textcolor{blue}{(-y)})(4+(-3x))\\ \textcolor{red}{2x}\cdot 4+\textcolor{red}{2x}\cdot(-3x)+\textcolor{blue}{(-y)}\cdot 4+\textcolor{blue}{(-y)} \cdot(-3x)\\ 8x-6x^2-4y+3xy$In the operations above we used the sign multiplication laws, and the exponent law for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we will combine similar terms. We will define similar terms as terms in which the variables, 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 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),

Note 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.

We will settle for arranging it again from the highest power to the lowest from left to right:
$\textcolor{purple}{ 8x}\textcolor{green}{-6x^2}-4y\textcolor{orange}{+3xy}\\ \textcolor{green}{-6x^2}\textcolor{orange}{+3xy}\textcolor{purple}{ +8x}-4y\\$We highlighted the different terms using colors, and as already emphasized before, we made sure that the sign before the term is correct.

We thus received that the correct answer is answer D.

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∗2a∗b∗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

We may use the parenthesis on the right hand side due to the fact that it can be simplified as follows :

(8+a)

Resulting in the following calculation:

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

$136+17a+8c+ca$

We begin by opening the parentheses using the distributive property in order to simplify the expression:

$x(y+z)=xy+xz$Note that in the distributive property formula we assume that there is addition between the terms inside of the parentheses, therefore it is crucial to take into account the sign of the coefficient of the term.

Furthermore, we apply the rules of multiplication of signs in order to present any expression within the parentheses. The parentheses are opened with the help of the distributive property, as an expression in which there is an addition operation between all the terms:

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

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