Extended Distributive Property: Applying the formula

Let's simplify the given expression by opening the parentheses using the extended distribution 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 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:

$(\textcolor{red}{x}+\textcolor{blue}{4})(x+3)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot3+\textcolor{blue}{4}\cdot x +\textcolor{blue}{4}\cdot3\\ x^2+3x+4x+12$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

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):
$\textcolor{purple}{x^2}\textcolor{green}{+3x}\textcolor{green}{+4x}+12\\ \textcolor{purple}{x^2}\textcolor{green}{+7x}+12$In 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,

Thus the correct answer is C.

Let's simplify the given expression, using the extended distribution law to open the parentheses :

$(\textcolor{red}{t}+\textcolor{blue}{b})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. Remember that the sign preceding the term is an inseparable part of it. Apply the rules of sign multiplication so that we can present any expression in parentheses. We'll open the parentheses using the above formula, as an expression where addition operation exists between all terms. In this expression it's clear, all terms have a plus sign prefix.

Therefore we'll proceed directly to opening the parentheses as shown below:

$$:

$(\textcolor{red}{a}+\textcolor{blue}{15})(5+a)\\ \textcolor{red}{a}\cdot 5+\textcolor{red}{a}\cdot a+\textcolor{blue}{15}\cdot 5 +\textcolor{blue}{15}\cdot a\\ 5a+a^2+75+15a$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

In the next step we'll combine like terms, which we define as terms where the variable (or variables each separately), in this case a, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is due to the fact that any number raised to the power of zero equals 1) Apply the commutative law of addition and arrange the expression from highest to lowest power from left to right (we'll treat the free number as power of zero):
$\textcolor{purple}{5a}\textcolor{green}{+a^2}+75\textcolor{purple}{+15a}\\ \textcolor{green}{a^2}\textcolor{purple}{+5a+15a}+75\\ \textcolor{green}{a^2}\textcolor{purple}{+20a}+75\\$In 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,

We therefore got that the correct answer is answer B.

Let's simplify the given expression,using the extended distribution law to open the parentheses :

$(\textcolor{red}{t}+\textcolor{blue}{k})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{k}c+\textcolor{blue}{k}d$

Note that in the formula template for the above distribution law, we take as a default that the operation between terms inside of the parentheses is addition. Remember that the sign preceding the term is an inseparable part of it. Apply the rules of sign multiplication and we can present any expression inside of the parentheses. We'll open the parentheses by using the above formula, as an expression where addition operation exists between all terms:

$(12-x)(x-3) \\ (\textcolor{red}{12}+\textcolor{blue}{(-x)})(x+(-3))\\$Let's begin then with opening the parentheses:

$(\textcolor{red}{12}+\textcolor{blue}{(-x)})(x+(-3))\\ \textcolor{red}{12}\cdot x+\textcolor{red}{12}\cdot(-3)+\textcolor{blue}{(-x)}\cdot x +\textcolor{blue}{(-x)}\cdot(-3)\\ 12x-36-x^2 +3x$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

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 zero power yields 1) Apply the commutative property of addition and proceed to arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{12x}\textcolor{green}{-36}-x^2\textcolor{purple}{+3x}\\ -x^2\textcolor{purple}{+12x+3x}\textcolor{green}{-36}\\ -x^2\textcolor{purple}{+15x}\textcolor{green}{-36}\\$In 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 remained an inseparable part of it,

We therefore got that the correct answer is answer A (we used the commutative property of addition to verify this).

In order to solve the given problem, we will use the FOIL method. FOIL stands for First, Outer, Inner, Last. This helps us to systematically expand the product of the two binomials:

• Step 1: Multiply the First terms.

The first terms of each binomial are $x$ and $x$. Multiply these together to obtain $x \times x = x^2$.

• Step 2: Multiply the Outer terms.

The outer terms are $x$ and $-4$. Multiply these. together to obtain $x \times -4 = -4x$.

• Step 3: Multiply the Inner terms.

The inner terms are $2$ and $x$. Multiply these together to obtain $2 \times x = 2x$.

• Step 4: Multiply the Last terms.

The last terms are $2$ and $-4$. Multiply these together to obtain $2 \times -4 = -8$.

Proceed to combine all these results together:

$x^2 - 4x + 2x - 8$

Finally, combine like terms:

Combine $-4x$ and $2x$ to obtain $-2x$.

The expanded form of the expression is therefore:

$x^2 - 2x - 8$

Thus, the solution to the problem is $x^2 - 2x - 8$, which corresponds to choice 1.

In order to simplify the given expression, open the parentheses using the extended distribution 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 in the formula template for the above distribution law, it is a given that the operation between the terms inside of the parentheses is addition. Furthermore the sign preceding the term is of great significance and must be taken into consideration;

Proceed to apply the above formula to the expression to open out the parentheses.

$(x-6)(x+8)\\ \downarrow\\ \big(\textcolor{red}{x}+\textcolor{blue}{(-6)}\big)(x+8)\\$Let's begin then with opening the parentheses:

$\big(\textcolor{red}{x}+\textcolor{blue}{(-6)}\big)(x+8)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot8+\textcolor{blue}{(-6)}\cdot x+\textcolor{blue}{(-6)}\cdot8\\ x^2+8x-6x-48$

To calculate the above multiplications operations we used the multiplication table as well as the laws of exponents for multiplication between terms with identical bases:

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

In the next step we'll combine like terms which we define as terms where the variable (or variables ), in this case x, have identical exponents . (Note that in the absence of one of the variables from the expression, we'll consider its exponent as zero power due to the fact that raising any number to the zero power yields the result 1) Apply the commutative property of addition and proceed to arrange the expression from highest to lowest power from left to right (Note: treat the free number as having zero power):
$\textcolor{purple}{x^2}\textcolor{green}{+8x-6x}\textcolor{orange}{-48}\\ \textcolor{purple}{x^2}\textcolor{green}{+2x}\textcolor{orange}{-48}\\$When combining like terms as shown above, we highlighted the different terms using colors, as well as treating the sign preceding the term as an inseparable part of it.

The correct answer is answer A.

Let's simplify the given expression, using the expanded distribution law in order to open the parentheses :

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. We must remember that the sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression inside of the parentheses. We'll open the parentheses using the above formula, first as an expression where addition operation exists between all terms:

$(x-8)(x+y)\\ (\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\$Proceed to open the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot y+\textcolor{blue}{(-8)}\cdot x +\textcolor{blue}{(-8)}\cdot y\\ x^2+xy-8x -8y$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

Note that in the expression that we obtained in the last stage there are four different terms, this is due to the fact that there isn't even one pair of terms where the variables (different ones) have the same exponent. Additionally the expression is already organized therefore the expression that we obtain is the final and most simplified form:
$\textcolor{purple}{ x^2}\textcolor{green}{+xy}-8x \textcolor{orange}{-8y}\\$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,

Therefore the correct answer is answer A.

To solve this problem, we'll apply the FOIL method for multiplying binomials:

• First: Multiply the first terms in each binomial: $(2x)(x) = 2x^2$.
• Outer: Multiply the outer terms in the product: $(2x)(3) = 6x$.
• Inner: Multiply the inner terms: $(y)(x) = xy$.
• Last: Multiply the last terms: $(y)(3) = 3y$.

Next, we combine these results to form the expanded expression:

$2x^2 + 6x + xy + 3y$.

Since terms $6x$ and $xy$ are not like terms, they cannot be combined, resulting in the final expression:

$2x^2 + xy + 6x + 3y$.

Upon reviewing the multiple-choice options, the correct answer is the expanded expression, choice 4: $2x^2 + xy + 6x + 3y$.

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:

$350+175=525$

$525+40=565$

$565+20=585$

To solve this problem, we'll use the distributive property, also known as the FOIL method when dealing with two binomials.

Let's expand the expression $(7+b)(a+9)$:

• First, apply the distributive property by multiplying each term in the first binomial by each term in the second binomial. This means we will have four operations:
• Step 1: Multiply $7$ by $a$. This gives $7a$.
• Step 2: Multiply $7$ by $9$. This gives $63$.
• Step 3: Multiply $b$ by $a$. This gives $ab$.
• Step 4: Multiply $b$ by $9$. This gives $9b$.

After performing these operations, the expression expands to:

$7a + 63 + ab + 9b$

Rearrange the terms in standard form for the final answer, which is:

$ab + 7a + 9b + 63$

Therefore, the solution to the problem is $ab + 7a + 9b + 63$.

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!

Let's simplify the expression by opening 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 (a).

To solve this problem, we'll perform a step-by-step expansion of the expression $(x+13)(y+4)$ using the distributive property:

• Step 1: Multiply the first terms $(x \cdot y) = xy$.
• Step 2: Multiply the outer terms $(x \cdot 4) = 4x$.
• Step 3: Multiply the inner terms $(13 \cdot y) = 13y$.
• Step 4: Multiply the last terms $(13 \cdot 4) = 52$.

After completing these steps, combine the results:

$xy + 4x + 13y + 52$

This is the final expanded form of the expression. By comparing with the given choices, the correct answer is:

$xy + 4x + 13y + 52$

Therefore, the correct choice is option 3.

To solve this problem, we need to multiply the binomials $(x-6)$ and $(x+2)$ using the distributive property (FOIL method):

• First: Multiply the first terms: $x \cdot x = x^2$
• Outer: Multiply the outer terms: $x \cdot 2 = 2x$
• Inner: Multiply the inner terms: $-6 \cdot x = -6x$
• Last: Multiply the last terms: $-6 \cdot 2 = -12$

Now, we have the terms: $x^2$, $2x$, $-6x$, and $-12$.
We combine the linear terms:

$x^2 + 2x - 6x - 12 = x^2 - 4x - 12$

This is the expanded form of the quadratic expression in standard form.
Therefore, the solution to the problem is $x^2 - 4x - 12$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the structure of the expression
• Step 2: Apply the difference of squares formula
• Step 3: Simplify the expression

Now, let's work through each step:
Step 1: The expression is $(x+y)(x-y)$, which resembles the difference of squares. Step 2: Using the formula for the difference of squares, $(a+b)(a-b) = a^2 - b^2$, we set $a = x$ and $b = y$. Step 3: Applying the formula, we have:

$(x+y)(x-y) = x^2 - y^2$.

Therefore, the solution to the problem is $x^2-y^2$.

Let's simplify the given expression and open the parentheses using the extended distribution 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 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 the sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(2x+3)(-5-x)\\ (\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\$Let's begin by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\ \textcolor{red}{2x}\cdot (-5)+\textcolor{red}{2x}\cdot(-x)+\textcolor{blue}{3}\cdot (-5) +\textcolor{blue}{3} \cdot(-x)\\ -10x-2x^2-15-3x$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

In the next step we'll combine like terms. Like terms are 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, due to the fact that raising any number to the power of zero yields the result 1) We'll apply the commutative law of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{-10x}\textcolor{green}{-2x^2}-15\textcolor{purple}{-3x}\\ \textcolor{green}{-2x^2} \textcolor{purple}{-10x}\textcolor{purple}{-3x}-15\\ \textcolor{green}{-2x^2}\textcolor{purple}{-13x}-15$

In 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 remains an inseparable part of it,

We therefore got that the correct answer is answer D.

Let's simplify the given expression by opening the parentheses using the extended distribution 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 in the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. We must remember that the sign preceding the term is an inseparable part of it. We will also apply the rules of sign multiplication in order to present any expression in the parentheses, which we will open using the above formula. First as an expression where 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 to open the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{y})(3x+2y)\\ \textcolor{red}{x}\cdot 3x+\textcolor{red}{x}\cdot2y+\textcolor{blue}{y}\cdot 3x+\textcolor{blue}{y} \cdot2y\\ 3x^2+2xy+3xy+2y^2$

In calculating the above multiplications, we used the multiplication table as well as the laws of exponents for multiplication between terms with identical bases:

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

In the next step we'll combine like terms, which we'll define as terms where the variable (or variables, each separately), in this case x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll treat its exponent as zero power. This is due to the fact that raising any number to the zero power yields the result 1) We'll apply the commutative law of addition, and proceed to arrange the expression (if needed) from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{ 3x^2}\textcolor{green}{+2xy}\textcolor{green}{+3xy}\textcolor{orange}{+2y^2}\\ \textcolor{purple}{ 3x^2}\textcolor{green}{+5xy}\textcolor{orange}{+2y^2}\\$We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term remains an inseparable part of it,

We therefore concluded that the correct answer is answer A.

Let's simplify the given expression, using the extended distribution law to open the parentheses :

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition. Remember that the sign preceding the term is an inseparable part of it. By applying the laws of sign multiplication we can present any expression inside of the parentheses. We 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 directly proceed to opening the parentheses,

$$Start by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3y})(2z+12m)\\ \textcolor{red}{2x}\cdot 2z+\textcolor{red}{2x}\cdot12m+\textcolor{blue}{3y}\cdot 2z+\textcolor{blue}{3y} \cdot12m\\ 4xz+24xm+6yz+36ym$

In the next step we'll combine like terms, which we define as terms where the variable (or variables, each separately), in this case z,m x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is due to the fact that any number raised to the power of zero equals 1),

Note that in the expression that we obtained in the last step there are four different terms. This is due to the fact that there isn't even one pair of terms where the (different) variables have the same exponent. Therefore the expression that we already obtained is the final and most simplified expression:
$\textcolor{purple}{ 4xz}\textcolor{green}{+24xm}\textcolor{black}{+6yz}\textcolor{orange}{+36ym }\\$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,

We can conclude that the correct answer is answer D.

Let's simplify the given expression by opening the parentheses using the extended distribution law:

$(\textcolor{red}{t}+\textcolor{blue}{k})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{k}c+\textcolor{blue}{k}d$

Note that in the formula template for the above distribution law, we take as a default that the operation between the terms inside the parentheses is addition. The sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression inside of the parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(3b+7a)(-5a+2b) \\ (\textcolor{red}{3b}+\textcolor{blue}{7a})((-5a)+2b)\\$Let's begin by opening the parentheses:

$(\textcolor{red}{3b}+\textcolor{blue}{7a})((-5a)+2b)\\ \textcolor{red}{3b}\cdot (-5a)+\textcolor{red}{3b}\cdot2b+\textcolor{blue}{7a}\cdot (-5a) +\textcolor{blue}{7a}\cdot2b\\ -15ab+6b^2-35a^2+14ab$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

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

In the next step, we'll combine like terms, which we define as terms where the variable(s) (or each variable separately), in this case an and b, 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 zero power yields 1) We'll apply the commutative law of addition as well as arrange the expression from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{-15ab}\textcolor{green}{+6b^2}-35a^2\textcolor{purple}{+14ab}\\ \textcolor{green}{6b^2}-35a^2\textcolor{purple}{-15ab}\textcolor{purple}{+14ab}\\ \textcolor{green}{6b^2}-35a^2\textcolor{purple}{-ab}$

In 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,

The correct answer is answer B.

In order to solve the given problem, we'll follow these steps:

• Step 1: Use the distributive property to expand the expression.

• Step 2: Combine like terms to simplify the expression.

Let's proceed to work through each step:

Step 1: Begin by applying the distributive property to expand the expression:

$(a-5b)(7a+3b) = a(7a) + a(3b) - 5b(7a) - 5b(3b)$

Calculate each term:

• $a \cdot 7a = 7a^2$

• $a \cdot 3b = 3ab$

• $-5b \cdot 7a = -35ab$

• $-5b \cdot 3b = -15b^2$

Merge together, as follows:

$7a^2 + 3ab - 35ab - 15b^2$

Step 2: Combine like terms:

The terms $3ab$ and $-35ab$ are like terms, hence we combine them:

$7a^2 + (3ab - 35ab) - 15b^2 = 7a^2 - 32ab - 15b^2$

Therefore, the solution to the problem is $7a^2 - 32ab - 15b^2$.

In order to simplify the given expression we must use the expanded distributive law as seen below:

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

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 whilst taking into account the correct multiplication of signs:

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