Extended Distributive Property: Matching expressions equal in value

To solve this problem, we need to expand each given algebraic expression using the distributive property and match each expanded form with the given standard forms. Let’s go through each expression:

Expression 1: $(b+c)(a-4)$

• Apply distributive property: $ba - 4b + ca - 4c$
• Combine like terms, resulting in: $ab + ac - 4b - 4c$.
• This matches with option a: $ac + ab - 4b - 4c$.

Expression 2: $(4+c)(a+b)$

• Apply distributive property: $4a + 4b + ca + cb$
• Combine like terms, resulting in: $4a + 4b + ac + cb$.
• This matches with option d: $4a + 4b + ac + cb$.

Expression 3: $(a+4)(b-c)$

• Apply distributive property: $ab - ac + 4b - 4c$
• Combine like terms, resulting in: $ab - ac + 4b - 4c$.
• This matches with option b: $4b + ab - 4c - ac$.

Expression 4: $(b+4)(c-a)$

• Apply distributive property: $bc - ba + 4c - 4a$
• Combine like terms, resulting in: $bc - ab + 4c - 4a$.
• This matches with option c: $bc - ab + 4c - 4a$.

Grouping the results, we have:

• 1 matches with a.
• 2 matches with d.
• 3 matches with b.
• 4 matches with c.

Therefore, the solution to the problem is 1-a, 2-d, 3-b, 4-c.

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

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$ 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. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(x+4)(x-3) \\ \downarrow\\ (x+4)\big(x+(-3)\big) \\ x\cdot x+x\cdot(-3)+4\cdot x+4\cdot(-3)\\ x^2-3x+4x-12\\ \boxed{x^2+x-12}$

2. $(x+4)(x+3) \\ x\cdot x+x\cdot 3+4\cdot x+4\cdot 3\\ x^2+3x+4x+12\\ \boxed{x^2+7x+12}$

3. $(x-4)(x-3) \\ \downarrow\\ \big(x+(-4)\big)\big(x+(-3)\big) \\ x\cdot x+x\cdot(-3)+(-4)\cdot x+(-4)\cdot(-3)\\ x^2-3x-4x+12\\ \boxed{x^2-7x+12}$

   After applying the commutative law of addition and multiplication we observe that:

   The simplified expression in 1 matches the expression in option A,

   The simplified expression in 2 matches the expression in option C,

   The simplified expression in 3 matches the expression in option B,

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

Let's simplify the given expressions, open the parentheses using the expanded distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$ 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. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(2x+y)(x+2y) \\ 2x\cdot x+2x\cdot 2y+y\cdot x+y\cdot2y\\ 2x^2+4xy+xy+2y^2\\ \boxed{2x^2+5xy+2y^2}$

2. $(2x+2y)(x+y) \\ 2x\cdot x+2x\cdot y+2y\cdot x+2y\cdot y\\ 2x^2+2xy+2yx+2y^2\\ \boxed{2x^2+4xy+2y^2}$

3. $(2x-y)(x-2y) \\ \downarrow\\ \big(2x+(-y)\big)\big(x+(-2y)\big) \\ 2x\cdot x+2x\cdot(-2y)+(-y)\cdot x+(-y)\cdot(-2y)\\ 2x^2-4xy-yx+2y^2\\ \boxed{2x^2-5xy+2y^2}$

   Now, we'll use the commutative law of addition and multiplication to notice that:

   The simplified expression in 1 matches the expression in option C,

   The simplified expression in 2 matches the expression in option A,

   The simplified expression in 3 matches the expression in option B,

Therefore, the correct answer (among the suggested options) is answer A.

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

$(\textcolor{red}{x}+\textcolor{blue}{y})(m+r)=\textcolor{red}{x}m+\textcolor{red}{x}r+\textcolor{blue}{y}m+\textcolor{blue}{y}r$

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. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(a+b)(c+d) \\ \boxed{ac+ad+bc+bd}\\$

2. $(a+c)(b+d) \\ \boxed{ab+ad+cb+cd}$

3. $(a+d)(c+b) \\ \boxed{ac+ab+dc+db}$

   In all expressions where we performed a multiplication operation between the expressions in the above parentheses, the result of the multiplication (obtained after applying the mentioned distribution law) yielded an expression where the terms cannot be combined. This is due to the fact that all terms in the resulting expression are different from each other ( All variables in like terms need to be identical and have the same exponent)

4. After applying the commutative law of addition and multiplication we observe that:

   The simplified expression in 1 matches the expression in option A,

   The simplified expression in 2 matches the expression in option C,

   The simplified expression in 3 matches the expression in option B,

Therefore the correct answer (among the suggested options) is answer C.

To solve this problem, we'll use the distributive property to expand each expression and find its equivalent form.

**Step 1: Expand each given expression.**

• For $(m-n)(a-4)$:

$(m-n)(a-4) = m \cdot a - m \cdot 4 - n \cdot a + n \cdot 4 = ma - 4m - na + 4n$

• For $(4-n)(m+a)$:

$(4-n)(m+a) = 4 \cdot m + 4 \cdot a - n \cdot m - n \cdot a = 4m + 4a - nm - na$

• For $(n-m)(4-a)$:

$(n-m)(4-a) = n \cdot 4 - n \cdot a - m \cdot 4 + m \cdot a = 4n - na - 4m + ma = -ma + 4m + na - 4n$

**Step 2: Match expanded expressions with the given options.**

• $ma - 4m - na + 4n$ matches option b.

• $4m + 4a - nm - na$ matches option a.

• $-ma + 4m + na - 4n$ matches option c.

The expanded expressions match the options as follows:

• Expression $(m-n)(a-4)$ matches option b.

• Expression $(4-n)(m+a)$ matches option a.

• Expression $(n-m)(4-a)$ matches option c.

Thus, the correct matches are 1-b, 2-a, 3-c.

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

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$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. $(2x-y)(x+3) \\ \downarrow\\ \big(2x+(-y)\big)(x+3) \\ 2x\cdot x+2x\cdot 3+(-y)\cdot x+(-y)\cdot3\\ \boxed{2x^2+6x-yx-3y}\\$

2. $(y-2x)(3-x) \\ \downarrow\\ \big(y+(-2x)\big)\big(3+(-x)\big) \\ y\cdot 3+y\cdot (-x)+(-2x)\cdot 3+(-2x)\cdot(-x)\\ \boxed{3y-xy-6x+2x^2}\\$

3. $(2x+y)(x-3) \\ \downarrow\\ (2x+y)(x+(-3)) \\ 2x\cdot x+2x\cdot (-3)+y\cdot x+y\cdot(-3)\\ \boxed{2x^2-6x+yx-3y}\\$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.

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

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

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. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(x+6)(x+8) \\ x\cdot x+x\cdot 8+6\cdot x+6\cdot8\\ x^2+8x+6x+48\\ \boxed{x^2+14x+48}\\$

2. $(6+x)(8-x) \\ \downarrow\\ (6+x)\big(8+(-x)\big) \\ 6\cdot 8+6\cdot (-x)+x\cdot 8+x\cdot(-x)\\ 48-6x+8x-x^2\\ \boxed{48+2x-x^2}\\$

3. $(x+x)(6+8) \\ 2x\cdot14\\ \boxed{28x}$

   In the last expression we simplified above, we first combined like terms in each of the expressions within parentheses, therefore in this case there was no need to use the extended distribution law mentioned at the beginning of the solution to simplify the expression.

   After applying the commutative law of addition and multiplication we observe that:

   The simplified expression in 1 matches the expression in option C,

   The simplified expression in 2 matches the expression in option A,

   The simplified expression in 3 matches the expression in option B,

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

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

• Step 1: Expand each original expression using the distributive property.
• Step 2: Compare each expansion with the given rearranged expressions to find the correct match.

Step 1: Expand each original expression.
- For Expression 1: $(a+g)x+3$ expands to $ax + gx + 3$.
- For Expression 2: $(x+3)(a+g)$ expands to $xa + xg + 3a + 3g$.
- For Expression 3: $(a-g)x-3$ expands to $ax - gx - 3$.

Step 2: Match each expanded expression to the provided options:
- Expression 1: $ax + gx + 3$ matches with option b.
- Expression 2: $xa + xg + 3a + 3g$ matches with option a.
- Expression 3: $ax - gx - 3$ matches with option c.

Therefore, the correct mapping of expressions is:
1-b, 2-a, 3-c.

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

• Step 1: Expand each factored expression using the distributive property.
• Step 2: Match the expanded expressions with their corresponding forms.

Step 1: Expand the factored expressions:

- $(2x+9)(y+4)$:
Apply the distributive property:
$(2x)y + (2x)4 + 9y + 36 = 2xy + 8x + 9y + 36$.

- $(2y+9)(x+4)$:
Apply the distributive property:
$(2y)x + (2y)4 + 9x + 36 = 2xy + 8y + 9x + 36$.

- $(2x-9)(y-4)$:
Apply the distributive property:
$(2x)y - (2x)4 - 9y + 36 = 2xy - 8x - 9y + 36$.

- $(2x+9)(y-4)$:
Apply the distributive property:
$(2x)y - (2x)4 + 9y - 36 = 2xy - 8x + 9y - 36$.

Step 2: Match the expanded forms to their corresponding choices:

• 4-b: $(2x+9)(y-4) = 2xy - 8x + 9y - 36$
• 3-c: $(2x-9)(y-4) = 2xy - 8x - 9y + 36$
• 2-d: $(2y+9)(x+4) = 2xy + 8y + 9x + 36$
• 1-a: $(2x+9)(y+4) = 2xy + 8x + 9y + 36$

Therefore, the correct matching of expressions is 4-b, 3-c, 2-d, 1-a.

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

• Step 1: Expand each of the given expressions using the distributive property.
• Step 2: Simplify the resulting expressions.
• Step 3: Match the simplified expressions with options a, b, and c.

Now, let's work through each step:
Step 1: Start with the first expression $3(y+b) + 4x$.
Applying the distributive property: $3 \cdot y + 3 \cdot b + 4x = 3y + 3b + 4x$. This matches with option a: $3y+3b+4x$.

Step 2: Consider the second expression $(3+4x)(y+b)$.
Expanding using the distributive property, we get: $3(y+b) + 4x(y+b) = 3y + 3b + 4xy + 4xb$. This matches with option c: $3y+3b+4xy+4xb$.

Step 3: Finally, expand the third expression $(4y+3)(x+b)$.
Apply the distributive property: $4y(x+b) + 3(x+b) = 4yx + 4yb + 3x + 3b$. This matches with option b: $4yx+4yb+3x+3b$.

Therefore, the matches are:
First expression matches option a
Second expression matches option c
Third expression matches option b

Therefore, the solution to the problem is 1-a, 2-c, 3-b.

To solve this problem, we'll match each bracketed pair of algebraic terms with its equivalent expanded form using the distributive property.

Step-by-Step Solution:

Expression 1: $(y+5)(x+7)$

• Apply FOIL method:
• First: $y \cdot x = xy$
  Outside: $y \cdot 7 = 7y$
  Inside: $5 \cdot x = 5x$
  Last: $5 \cdot 7 = 35$
• Combine: $xy + 7y + 5x + 35$
• Match: Option a $(xy + 7y + 5x + 35)$

Expression 2: $(x+5)(y+7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot 7 = 7x$
  Inside: $5 \cdot y = 5y$
  Last: $5 \cdot 7 = 35$
• Combine: $xy + 7x + 5y + 35$
• Match: Option b $(xy + 7x + 5y + 35)$

Expression 3: $(x-5)(y-7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot (-7) = -7x$
  Inside: $(-5) \cdot y = -5y$
  Last: $(-5) \cdot (-7) = 35$
• Combine: $xy - 7x - 5y + 35$
• Match: Option c $(xy - 7x - 5y + 35)$

Expression 4: $(x-5)(y+7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot 7 = 7x$
  Inside: $(-5) \cdot y = -5y$
  Last: $(-5) \cdot 7 = -35$
• Combine: $xy + 7x - 5y - 35$
• Match: Option d $(xy + 7x - 5y - 35)$

By matching each expression with its expanded equivalent, we conclude:

1-a, 2-b, 3-c, 4-d

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

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

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. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(12x-5)(y+2) \\ \downarrow\\ \big(12x+(-5)\big)(y+2)\\ 12x\cdot y+12x\cdot 2+(-5)\cdot y+(-5)\cdot 2\\ \boxed{12xy+24x-5y-10}\\$

2. $(x-12)(5y+2) \\ \downarrow\\ \big(x+(-12)\big)(5y+2)\\ x\cdot 5y+x\cdot 2+(-12)\cdot 5y+(-12)\cdot 2\\ \boxed{5xy+2x-60y-24}\\$

3. $(12x+5)(y-2) \\ \downarrow\\ (12x+5)\big(y+(-2)\big)\\ 12x\cdot y+12x\cdot (-2)+5\cdot y+5\cdot (-2)\\\ \boxed{12xy-24x+5y-10}\\$

   Note in all expressions where we performed the multiplication between the expressions in the parentheses above, the result of the multiplication (obtained after applying the aforementioned distribution law) yielded an expression where terms cannot be combined. Due to the fact that all terms in the resulting expression are different from each other (remember that all variables in like terms need to be identical and have the same exponent),

   After applying the commutative law of addition and multiplication we observe that:

   The simplified expression in 1 matches the expression in option B,

   The simplified expression in 2 matches the expression in option C,

   The simplified expression in 3 matches the expression in option A,

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

To solve this problem, we'll expand each expression and find which polynomial they correspond with:

Start with expression 1: $(4+x)(y+8+x)$.

• Using the distributive property, expand as follows:
  $= 4(y+8+x) + x(y+8+x)$
  $= 4y + 32 + 4x + xy + 8x + x^2$
  $= x^2 + xy + 12x + 4y + 32$

This matches the polynomial $x^2 + 12x + xy + 4y + 32$, which is option b.

Next, consider expression 2: $(4+x+y)(8+x)$.

• Once again, expand using distributive property:
  $= (4+x+y)8 + (4+x+y)x$
  $= 32 + 8x + 8y + 4x + x^2 + xy$
  $= x^2 + 12x + xy + 8y + 32$

This matches the polynomial $x^2 + 12x + xy + 8y + 32$, which is option c.

Finally, consider expression 3: $(12+x)(y+x)$.

• Use distributive property to expand:
  $= 12(y+x) + x(y+x)$
  $= 12y + 12x + xy + x^2$
  $= x^2 + 12x + xy + 12y$

This matches the polynomial $x^2 + 12x + xy + 12y$, which is option a.

The correct matches are therefore: 1-c, 2-b, 3-a.

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

• Step 1: Expand each given expression using the distributive property.
• Step 2: Simplify the expressions obtained from expansion.
• Step 3: Compare the simplified expressions to the given expanded forms (a, b, and c).
• Step 4: Determine which expanded forms correspond to which original expressions and select the correct answer choice.

Let's begin with Step 1:
Expression 1: $(-7-x)(a-13)$

Expand:
$(-7)(a) + (-7)(-13) + (-x)(a) + (-x)(-13)$ = $-7a + 91 - ax + 13x$

Expression 2: $(-a+13)(-7-x)$

Expand:
$(-a)(-7) + (-a)(-x) + 13(-7) + 13(-x)$ = $7a + ax - 91 - 13x$

Expression 3: $(7+x)(a-13)$

Expand:
$7a - 91 + ax - 13x$

Now proceed with Step 3: Comparing these expanded expressions to the provided forms:

• Expression 1: $-7a + 91 - ax + 13x$ corresponds with Form b: $-7a + 91 - ax + 13x$.
• Expression 2: $7a + ax - 91 - 13x$ matches with Form a: $7a + ax - 91 - 13x$.
• Expression 3: $7a - 91 + ax - 13x$ is equivalent to Form a: $7a + ax - 91 - 13x$ after rearranging terms.

After comparing, we obtain the answer selection as 1-b, 2,3-a.

To solve this problem, we need to expand each given expression and compare it to the list of provided expanded expressions.

Let's expand each of the four expressions:

• Expression 1: $(2a+b)(b+4)$

Applying the distributive property:

$(2a+b)(b) + (2a+b)(4) = 2ab + b^2 + 8a + 4b$ This matches the expanded form $2ab + b^2 + 8a + 4b$, option d.
• Expression 2: $(4+a)(2b+b) = (4+a)(3b)$

Distributing the terms:

$3b \cdot 4 + 3b \cdot a = 12b + 3ab$ This matches the expanded form $12b + 3ab$, option b.
• Expression 3: $(2a-b)(b-4)$

Distributing the terms:

$2a \cdot b - 2a \cdot 4 - b \cdot b + b \cdot 4 = 2ab - 8a - b^2 + 4b$ This matches the expanded form $2ab - 8a - b^2 + 4b$, option a.
• Expression 4: $(2a-b)(b+4)$

Distributing the terms:

$2a \cdot b + 2a \cdot 4 - b \cdot b - b \cdot 4 = 2ab + 8a - b^2 - 4b$ This matches the expanded form $2ab + 8a - b^2 - 4b$, option c.

Therefore, the correct matching is as follows:

• Expression 1: $(2a+b)(b+4)$ matches with $2ab+8a+b^2+4b$
• Expression 2: $(4+a)(2b+b)$ matches with $12b+3ab$
• Expression 3: $(2a-b)(b-4)$ matches with $2ab-8a-b^2+4b$
• Expression 4: $(2a-b)(b+4)$ matches with $2ab+8a-b^2-4b$

Thus, the correct answer is: 1-d, 2-b, 3-a, 4-c.

We use all the exercises of the extended distributive property:$(a+b)\times(c+d)=ac+ad+bc+bd$

1.$(a-b)(c-4)=ac-4a-bc+4b$

2.$(a+b)(c+4)=ac+4a+bc+4b$