Asked by jimmy
What is the difference between the expression (3x+4x)+2x
(
3
𝑥
+
4
𝑥
)
+
2
𝑥
and the expression 3x+(4x+2x)
3
𝑥
+
(
4
𝑥
+
2
𝑥
)
?(1 point)
Responses
2x
2
𝑥
2 x
3x
3
𝑥
3 x
4x
4
𝑥
4 x
0
(
3
𝑥
+
4
𝑥
)
+
2
𝑥
and the expression 3x+(4x+2x)
3
𝑥
+
(
4
𝑥
+
2
𝑥
)
?(1 point)
Responses
2x
2
𝑥
2 x
3x
3
𝑥
3 x
4x
4
𝑥
4 x
0
Answers
Answered by
jimmy
Which of the following equations demonstrates the Associative Property?(1 point)
Responses
3x⋅(4x⋅5)=3x(4x+5)
3
𝑥
⋅
(
4
𝑥
⋅
5
)
=
3
𝑥
(
4
𝑥
+
5
)
3 x times left parenthesis 4 x times 5 right parenthesis equals 3 x left parenthesis 4 x plus 5 right parenthesis
3x⋅(4x⋅5)=(3x⋅4x)⋅5
3
𝑥
⋅
(
4
𝑥
⋅
5
)
=
(
3
𝑥
⋅
4
𝑥
)
⋅
5
3 x times left parenthesis 4 x times 5 right parenthesis equals left parenthesis 3 x times 4 x right parenthesis times 5
(3x+4x)⋅5=3x+(4x⋅5)
(
3
𝑥
+
4
𝑥
)
⋅
5
=
3
𝑥
+
(
4
𝑥
⋅
5
)
left parenthesis 3 x plus 4 x right parenthesis times 5 equals 3 x plus left parenthesis 4 x times 5 right parenthesis
3x⋅(4x+5x)=(3x+4x)⋅5
Responses
3x⋅(4x⋅5)=3x(4x+5)
3
𝑥
⋅
(
4
𝑥
⋅
5
)
=
3
𝑥
(
4
𝑥
+
5
)
3 x times left parenthesis 4 x times 5 right parenthesis equals 3 x left parenthesis 4 x plus 5 right parenthesis
3x⋅(4x⋅5)=(3x⋅4x)⋅5
3
𝑥
⋅
(
4
𝑥
⋅
5
)
=
(
3
𝑥
⋅
4
𝑥
)
⋅
5
3 x times left parenthesis 4 x times 5 right parenthesis equals left parenthesis 3 x times 4 x right parenthesis times 5
(3x+4x)⋅5=3x+(4x⋅5)
(
3
𝑥
+
4
𝑥
)
⋅
5
=
3
𝑥
+
(
4
𝑥
⋅
5
)
left parenthesis 3 x plus 4 x right parenthesis times 5 equals 3 x plus left parenthesis 4 x times 5 right parenthesis
3x⋅(4x+5x)=(3x+4x)⋅5
Answered by
jimmy
Which expression is equivalent to (3r⋅5)⋅(−3⋅8)
(
3
𝑟
⋅
5
)
⋅
(
−
3
⋅
8
)
?(1 point)
Responses
−45r⋅5
−
45
𝑟
⋅
5
negative 45 r times 5
24r⋅(5⋅−3)
24
𝑟
⋅
(
5
⋅
−
3
)
24 r times left parenthesis 5 times negative 3 right parenthesis
15r⋅(5⋅8)
15
𝑟
⋅
(
5
⋅
8
)
15 r times left parenthesis 5 times 8 right parenthesis
(5⋅−3)⋅−9r
(
3
𝑟
⋅
5
)
⋅
(
−
3
⋅
8
)
?(1 point)
Responses
−45r⋅5
−
45
𝑟
⋅
5
negative 45 r times 5
24r⋅(5⋅−3)
24
𝑟
⋅
(
5
⋅
−
3
)
24 r times left parenthesis 5 times negative 3 right parenthesis
15r⋅(5⋅8)
15
𝑟
⋅
(
5
⋅
8
)
15 r times left parenthesis 5 times 8 right parenthesis
(5⋅−3)⋅−9r
Answered by
jimmy
Use the Distributive Property to rewrite the expression (3x−5)(2x+2)
(
3
𝑥
−
5
)
(
2
𝑥
+
2
)
.(1 point)
Responses
6x2−3x−10
6
𝑥
2
−
3
𝑥
−
10
6 x squared minus 3 x minus 10
x2−3x−10
𝑥
2
−
3
𝑥
−
10
x squared minus 3 x minus 10
6x2−4x−10
6
𝑥
2
−
4
𝑥
−
10
6 x squared minus 4 x minus 10
5x2−2x−3
(
3
𝑥
−
5
)
(
2
𝑥
+
2
)
.(1 point)
Responses
6x2−3x−10
6
𝑥
2
−
3
𝑥
−
10
6 x squared minus 3 x minus 10
x2−3x−10
𝑥
2
−
3
𝑥
−
10
x squared minus 3 x minus 10
6x2−4x−10
6
𝑥
2
−
4
𝑥
−
10
6 x squared minus 4 x minus 10
5x2−2x−3
Answered by
jimmy
Use the Distributive Property to write the expression 4x2−36
4
𝑥
2
−
36
a different way.(1 point)
Responses
(x+18)(x−18)
(
𝑥
+
18
)
(
𝑥
−
18
)
left parenthesis x plus 18 right parenthesis left parenthesis x minus 18 right parenthesis
4x(x−9)
4
𝑥
(
𝑥
−
9
)
4 x left parenthesis x minus 9 right parenthesis
(2x+9)(x−9)
(
2
𝑥
+
9
)
(
𝑥
−
9
)
left parenthesis 2 x plus 9 right parenthesis left parenthesis x minus 9 right parenthesis
(2x+6)(2x−6)
4
𝑥
2
−
36
a different way.(1 point)
Responses
(x+18)(x−18)
(
𝑥
+
18
)
(
𝑥
−
18
)
left parenthesis x plus 18 right parenthesis left parenthesis x minus 18 right parenthesis
4x(x−9)
4
𝑥
(
𝑥
−
9
)
4 x left parenthesis x minus 9 right parenthesis
(2x+9)(x−9)
(
2
𝑥
+
9
)
(
𝑥
−
9
)
left parenthesis 2 x plus 9 right parenthesis left parenthesis x minus 9 right parenthesis
(2x+6)(2x−6)
Answered by
jimmy
Find the expression that is equivalent to 3x2+2x−5x−2x2+2
3
𝑥
2
+
2
𝑥
−
5
𝑥
−
2
𝑥
2
+
2
.(1 point)
Responses
(x−2)(x−1)
(
𝑥
−
2
)
(
𝑥
−
1
)
left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis
(3x+2)(x+1)
(
3
𝑥
+
2
)
(
𝑥
+
1
)
left parenthesis 3 x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis
(x+1)(x+2)
(
𝑥
+
1
)
(
𝑥
+
2
)
left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis
(x+2)(x−2)
3
𝑥
2
+
2
𝑥
−
5
𝑥
−
2
𝑥
2
+
2
.(1 point)
Responses
(x−2)(x−1)
(
𝑥
−
2
)
(
𝑥
−
1
)
left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis
(3x+2)(x+1)
(
3
𝑥
+
2
)
(
𝑥
+
1
)
left parenthesis 3 x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis
(x+1)(x+2)
(
𝑥
+
1
)
(
𝑥
+
2
)
left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis
(x+2)(x−2)
Answered by
GPT-4o mini
To find the difference between the expressions \( (3x + 4x) + 2x \) and \( 3x + (4x + 2x) \), we can simplify both expressions.
1. For the first expression \( (3x + 4x) + 2x \):
- Simplify \( 3x + 4x \) to get \( 7x \).
- Then, \( 7x + 2x = 9x \).
2. For the second expression \( 3x + (4x + 2x) \):
- Simplify \( 4x + 2x \) to get \( 6x \).
- Then, \( 3x + 6x = 9x \).
Now, both expressions simplify to \( 9x \). Therefore, there is **no difference** between the two expressions.
The correct response is **0**.
1. For the first expression \( (3x + 4x) + 2x \):
- Simplify \( 3x + 4x \) to get \( 7x \).
- Then, \( 7x + 2x = 9x \).
2. For the second expression \( 3x + (4x + 2x) \):
- Simplify \( 4x + 2x \) to get \( 6x \).
- Then, \( 3x + 6x = 9x \).
Now, both expressions simplify to \( 9x \). Therefore, there is **no difference** between the two expressions.
The correct response is **0**.
Answered by
GPT-4o mini
The Associative Property states that the way in which numbers are grouped in addition or multiplication does not change their sum or product.
Given the options:
1. **\(3x \cdot (4x \cdot 5) = 3x(4x + 5)\)** - This equation does not demonstrate the Associative Property; it mixes multiplication and addition incorrectly.
2. **\(3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5\)** - This equation correctly shows the Associative Property of multiplication, where the grouping of factors can be changed without affecting the product.
3. **\((3x + 4x) \cdot 5 = 3x + (4x \cdot 5)\)** - This equation does not demonstrate the Associative Property; it does not correctly apply the property of addition or multiplication.
4. **\(3x \cdot (4x + 5x) = (3x + 4x) \cdot 5\)** - This mixes multiplication and addition incorrectly as well, so it does not demonstrate the Associative Property.
The correct response that demonstrates the Associative Property is:
**\(3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5\)**.
Given the options:
1. **\(3x \cdot (4x \cdot 5) = 3x(4x + 5)\)** - This equation does not demonstrate the Associative Property; it mixes multiplication and addition incorrectly.
2. **\(3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5\)** - This equation correctly shows the Associative Property of multiplication, where the grouping of factors can be changed without affecting the product.
3. **\((3x + 4x) \cdot 5 = 3x + (4x \cdot 5)\)** - This equation does not demonstrate the Associative Property; it does not correctly apply the property of addition or multiplication.
4. **\(3x \cdot (4x + 5x) = (3x + 4x) \cdot 5\)** - This mixes multiplication and addition incorrectly as well, so it does not demonstrate the Associative Property.
The correct response that demonstrates the Associative Property is:
**\(3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5\)**.
Answered by
GPT-4o mini
To simplify the expression \((3r \cdot 5) \cdot (-3 \cdot 8)\) and find an equivalent expression, we first calculate each part separately:
1. **Calculate \(3r \cdot 5\)**:
- This gives \(15r\).
2. **Calculate \(-3 \cdot 8\)**:
- This gives \(-24\).
Now we can combine these results:
\[
(3r \cdot 5) \cdot (-3 \cdot 8) = 15r \cdot (-24)
\]
This can be written as:
\[
15r \cdot -24 = -360r
\]
Let's check each answer choice to see if any match \(-360r\):
1. **−45r⋅5**:
- This simplifies to \(-225r\) (not equivalent).
2. **24r⋅(5⋅−3)**:
- This simplifies to \(24r \cdot (-15) = -360r\) (equivalent).
3. **15r⋅(5⋅8)**:
- This simplifies to \(15r \cdot 40 = 600r\) (not equivalent).
4. **(5⋅−3)⋅−9r**:
- This simplifies to \(-15 \cdot -9r = 135r\) (not equivalent).
The expression that is equivalent to \((3r \cdot 5) \cdot (-3 \cdot 8)\) is:
**24r⋅(5⋅−3)**.
1. **Calculate \(3r \cdot 5\)**:
- This gives \(15r\).
2. **Calculate \(-3 \cdot 8\)**:
- This gives \(-24\).
Now we can combine these results:
\[
(3r \cdot 5) \cdot (-3 \cdot 8) = 15r \cdot (-24)
\]
This can be written as:
\[
15r \cdot -24 = -360r
\]
Let's check each answer choice to see if any match \(-360r\):
1. **−45r⋅5**:
- This simplifies to \(-225r\) (not equivalent).
2. **24r⋅(5⋅−3)**:
- This simplifies to \(24r \cdot (-15) = -360r\) (equivalent).
3. **15r⋅(5⋅8)**:
- This simplifies to \(15r \cdot 40 = 600r\) (not equivalent).
4. **(5⋅−3)⋅−9r**:
- This simplifies to \(-15 \cdot -9r = 135r\) (not equivalent).
The expression that is equivalent to \((3r \cdot 5) \cdot (-3 \cdot 8)\) is:
**24r⋅(5⋅−3)**.
Answered by
GPT-4o mini
To rewrite the expression \((3x - 5)(2x + 2)\) using the Distributive Property, we will distribute each term in the first parenthesis by each term in the second parenthesis.
1. **Distributing \(3x\)**:
- \(3x \cdot 2x = 6x^2\)
- \(3x \cdot 2 = 6x\)
2. **Distributing \(-5\)**:
- \(-5 \cdot 2x = -10x\)
- \(-5 \cdot 2 = -10\)
Now, combine all these results:
\[
(3x - 5)(2x + 2) = 6x^2 + 6x - 10x - 10
\]
Combine like terms (\(6x - 10x\)):
\[
6x^2 - 4x - 10
\]
Thus, the expression rewritten is:
**6x² - 4x - 10**.
This corresponds to the third response option:
**6x² − 4x − 10**.
1. **Distributing \(3x\)**:
- \(3x \cdot 2x = 6x^2\)
- \(3x \cdot 2 = 6x\)
2. **Distributing \(-5\)**:
- \(-5 \cdot 2x = -10x\)
- \(-5 \cdot 2 = -10\)
Now, combine all these results:
\[
(3x - 5)(2x + 2) = 6x^2 + 6x - 10x - 10
\]
Combine like terms (\(6x - 10x\)):
\[
6x^2 - 4x - 10
\]
Thus, the expression rewritten is:
**6x² - 4x - 10**.
This corresponds to the third response option:
**6x² − 4x − 10**.
Answered by
GPT-4o mini
To rewrite the expression \(4x^2 - 36\) using the Distributive Property, we can first recognize that it can be factored as a difference of squares.
The expression \(4x^2 - 36\) can be rewritten as:
\[
4(x^2 - 9)
\]
Next, \(x^2 - 9\) is also a difference of squares and can be factored as:
\[
x^2 - 9 = (x + 3)(x - 3)
\]
Putting it all together, we can express \(4x^2 - 36\) as:
\[
4(x^2 - 9) = 4((x + 3)(x - 3)) = (2(x + 3))(2(x - 3)) = (2x + 6)(2x - 6)
\]
Now, let's compare this with the given options:
1. **\((x + 18)(x - 18)\)** - This is incorrect; it doesn't match our expression.
2. **\(4x(x - 9)\)** - This is also incorrect; it doesn't match our expression.
3. **\((2x + 9)(x - 9)\)** - This does not match our expression.
4. **\((2x + 6)(2x - 6)\)** - This matches our expression.
Thus, the correct answer is:
**\((2x + 6)(2x - 6)\)**.
The expression \(4x^2 - 36\) can be rewritten as:
\[
4(x^2 - 9)
\]
Next, \(x^2 - 9\) is also a difference of squares and can be factored as:
\[
x^2 - 9 = (x + 3)(x - 3)
\]
Putting it all together, we can express \(4x^2 - 36\) as:
\[
4(x^2 - 9) = 4((x + 3)(x - 3)) = (2(x + 3))(2(x - 3)) = (2x + 6)(2x - 6)
\]
Now, let's compare this with the given options:
1. **\((x + 18)(x - 18)\)** - This is incorrect; it doesn't match our expression.
2. **\(4x(x - 9)\)** - This is also incorrect; it doesn't match our expression.
3. **\((2x + 9)(x - 9)\)** - This does not match our expression.
4. **\((2x + 6)(2x - 6)\)** - This matches our expression.
Thus, the correct answer is:
**\((2x + 6)(2x - 6)\)**.
Answered by
GPT-4o mini
To find an expression equivalent to \(3x^2 + 2x - 5x - 2x^2 + 2\), we can first combine like terms:
1. Combine the \(x^2\) terms:
\[
3x^2 - 2x^2 = x^2
\]
2. Combine the \(x\) terms:
\[
2x - 5x = -3x
\]
3. The constant term is:
\[
+2
\]
Now we can rewrite the entire expression with the simplified terms:
\[
x^2 - 3x + 2
\]
Next, we will factor the quadratic expression \(x^2 - 3x + 2\):
We need two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-1\) and \(-2\) satisfy these conditions:
\[
(x - 1)(x - 2)
\]
Now let's compare this factored form with the given options:
1. **\((x - 2)(x - 1)\)** - This is equivalent to \(x^2 - 3x + 2\).
2. **\((3x + 2)(x + 1)\)** - This does not match our expression.
3. **\((x + 1)(x + 2)\)** - This does not match our expression.
4. **\((x + 2)(x - 2)\)** - This does not match our expression.
Thus, the correct answer is:
**\((x - 2)(x - 1)\)**.
1. Combine the \(x^2\) terms:
\[
3x^2 - 2x^2 = x^2
\]
2. Combine the \(x\) terms:
\[
2x - 5x = -3x
\]
3. The constant term is:
\[
+2
\]
Now we can rewrite the entire expression with the simplified terms:
\[
x^2 - 3x + 2
\]
Next, we will factor the quadratic expression \(x^2 - 3x + 2\):
We need two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-1\) and \(-2\) satisfy these conditions:
\[
(x - 1)(x - 2)
\]
Now let's compare this factored form with the given options:
1. **\((x - 2)(x - 1)\)** - This is equivalent to \(x^2 - 3x + 2\).
2. **\((3x + 2)(x + 1)\)** - This does not match our expression.
3. **\((x + 1)(x + 2)\)** - This does not match our expression.
4. **\((x + 2)(x - 2)\)** - This does not match our expression.
Thus, the correct answer is:
**\((x - 2)(x - 1)\)**.
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