Asked by x
According to the Distributive Property, which is a way to rewrite the algebraic expression (3x−1)(x−4) ?(1 point)
Responses
3x(−1)+3x(3x)+x(−4)+x(x)
3 x left parenthesis negative 1 right parenthesis plus 3 x left parenthesis 3 x right parenthesis plus x left parenthesis negative 4 right parenthesis plus x left parenthesis x right parenthesis
3x^2−5x+4
3 x squared minus 5 x plus 4
3x^2+6x+4
3 x squared plus 6 x plus 4
3x(x)+3x(−4)+−1(x)+−1(−4)
Responses
3x(−1)+3x(3x)+x(−4)+x(x)
3 x left parenthesis negative 1 right parenthesis plus 3 x left parenthesis 3 x right parenthesis plus x left parenthesis negative 4 right parenthesis plus x left parenthesis x right parenthesis
3x^2−5x+4
3 x squared minus 5 x plus 4
3x^2+6x+4
3 x squared plus 6 x plus 4
3x(x)+3x(−4)+−1(x)+−1(−4)
Answers
Answered by
x
Use the Distributive Property to find which expression is equivalent to 9x^2−25 .(1 point)
Responses
(3x+5)2
left parenthesis 3 x plus 5 right parenthesis squared
(3x+5)(3x−5)
left parenthesis 3 x plus 5 right parenthesis left parenthesis 3 x minus 5 right parenthesis
9x(x−25)
9 x left parenthesis x minus 25 right parenthesis
(4.5x+12.5)(4.5x−12.5)
Responses
(3x+5)2
left parenthesis 3 x plus 5 right parenthesis squared
(3x+5)(3x−5)
left parenthesis 3 x plus 5 right parenthesis left parenthesis 3 x minus 5 right parenthesis
9x(x−25)
9 x left parenthesis x minus 25 right parenthesis
(4.5x+12.5)(4.5x−12.5)
Answered by
x
Use the Distributive Property to verify which expression is equivalent to (x−5)(x−3) .(1 point)
Responses
x^2−8x+15
x squared minus 8 x plus 15
3x^2−4x^2−3x−15
3 x squared minus 4 x squared minus 3 x minus 15
x^2−2x+15
x squared minus 2 x plus 15
2x−2
Responses
x^2−8x+15
x squared minus 8 x plus 15
3x^2−4x^2−3x−15
3 x squared minus 4 x squared minus 3 x minus 15
x^2−2x+15
x squared minus 2 x plus 15
2x−2
Answered by
x
Which expression is equivalent to (z+4)(z−4) ?(1 point)
Responses
z^2+4z−4z−16
z squared plus 4 z minus 4 z minus 16
z(z)+4z(z)−4(4)
z left parenthesis z right parenthesis plus 4 z left parenthesis z right parenthesis minus 4 left parenthesis 4 right parenthesis
z^2−8z−16
z squared minus 8 z minus 16
2z^2−z2−8
Responses
z^2+4z−4z−16
z squared plus 4 z minus 4 z minus 16
z(z)+4z(z)−4(4)
z left parenthesis z right parenthesis plus 4 z left parenthesis z right parenthesis minus 4 left parenthesis 4 right parenthesis
z^2−8z−16
z squared minus 8 z minus 16
2z^2−z2−8
Answered by
x
What is another way to write the expression 2(x+1)(x+2) ?(1 point)
Responses
2x^2+5x+2
2 x squared plus 5 x plus 2
x^2+6x+2
x squared plus 6 x plus 2
2x^2+3x+2
2 x squared plus 3 x plus 2
2x^2+6x+4
Responses
2x^2+5x+2
2 x squared plus 5 x plus 2
x^2+6x+2
x squared plus 6 x plus 2
2x^2+3x+2
2 x squared plus 3 x plus 2
2x^2+6x+4
Answered by
GPT-4o mini
To rewrite the algebraic expression \( (3x - 1)(x - 4) \) using the Distributive Property, we will need to distribute each term in the first parenthesis by each term in the second parenthesis.
Let's do the step-by-step distribution:
1. **Distribute \( 3x \):**
- \( 3x \times x = 3x^2 \)
- \( 3x \times (-4) = -12x \)
2. **Distribute \( -1 \):**
- \( -1 \times x = -x \)
- \( -1 \times (-4) = 4 \)
Now, we combine all of these results together:
- \( 3x^2 - 12x - x + 4 \)
Next, combine the like terms:
- The terms \( -12x \) and \( -x \) combine to \( -13x \).
So the final expression becomes:
- \( 3x^2 - 13x + 4 \)
However, note that this expression does not match any options given, so let's check the expected responses closely:
1. \( 3x(−1)+3x(3x)+x(−4)+x(x) \) - this is not correctly simplified.
2. \( 3x^2−5x+4 \) - this does not match any calculations.
3. \( 3x^2+6x+4 \) - this is incorrect according to our calculations.
4. \( 3x(x)+3x(−4)+−1(x)+−1(−4) \) - this is a breakdown of the distribution, but it does not represent the simplified result.
Thus, it seems that none of these responses correctly represent the outcome of \( (3x - 1)(x - 4) \) after applying the Distributive Property correctly; however, option 4 shows the correct distribution steps before simplification.
Let's do the step-by-step distribution:
1. **Distribute \( 3x \):**
- \( 3x \times x = 3x^2 \)
- \( 3x \times (-4) = -12x \)
2. **Distribute \( -1 \):**
- \( -1 \times x = -x \)
- \( -1 \times (-4) = 4 \)
Now, we combine all of these results together:
- \( 3x^2 - 12x - x + 4 \)
Next, combine the like terms:
- The terms \( -12x \) and \( -x \) combine to \( -13x \).
So the final expression becomes:
- \( 3x^2 - 13x + 4 \)
However, note that this expression does not match any options given, so let's check the expected responses closely:
1. \( 3x(−1)+3x(3x)+x(−4)+x(x) \) - this is not correctly simplified.
2. \( 3x^2−5x+4 \) - this does not match any calculations.
3. \( 3x^2+6x+4 \) - this is incorrect according to our calculations.
4. \( 3x(x)+3x(−4)+−1(x)+−1(−4) \) - this is a breakdown of the distribution, but it does not represent the simplified result.
Thus, it seems that none of these responses correctly represent the outcome of \( (3x - 1)(x - 4) \) after applying the Distributive Property correctly; however, option 4 shows the correct distribution steps before simplification.
Answered by
GPT-4o mini
To determine which expression is equivalent to \( 9x^2 - 25 \) using the Distributive Property, we can recognize that \( 9x^2 - 25 \) is a difference of squares, which can be factored using the formula:
\[
a^2 - b^2 = (a + b)(a - b)
\]
In this case:
- \( a^2 = 9x^2 \), which gives us \( a = 3x \)
- \( b^2 = 25 \), which gives us \( b = 5 \)
Therefore, we can express \( 9x^2 - 25 \) as:
\[
9x^2 - 25 = (3x)^2 - (5)^2 = (3x + 5)(3x - 5)
\]
Now, let's analyze the provided responses:
1. **\( (3x + 5)^2 \)**: This expands to \( (3x + 5)(3x + 5) = 9x^2 + 30x + 25 \), which is not equivalent to \( 9x^2 - 25 \).
2. **\( (3x + 5)(3x - 5) \)**: This expands to \( 9x^2 - 25 \), which matches the original expression.
3. **\( 9x(x - 25) \)**: This expands to \( 9x^2 - 225x \), which is also not equivalent to \( 9x^2 - 25 \).
4. **\( (4.5x + 12.5)(4.5x - 12.5) \)**: This expands to a different expression. It calculates as \( (4.5x)^2 - (12.5)^2 = 20.25x^2 - 156.25 \), which does not match \( 9x^2 - 25 \).
Based on this analysis, the correct answer is:
**\( (3x + 5)(3x - 5) \)**.
\[
a^2 - b^2 = (a + b)(a - b)
\]
In this case:
- \( a^2 = 9x^2 \), which gives us \( a = 3x \)
- \( b^2 = 25 \), which gives us \( b = 5 \)
Therefore, we can express \( 9x^2 - 25 \) as:
\[
9x^2 - 25 = (3x)^2 - (5)^2 = (3x + 5)(3x - 5)
\]
Now, let's analyze the provided responses:
1. **\( (3x + 5)^2 \)**: This expands to \( (3x + 5)(3x + 5) = 9x^2 + 30x + 25 \), which is not equivalent to \( 9x^2 - 25 \).
2. **\( (3x + 5)(3x - 5) \)**: This expands to \( 9x^2 - 25 \), which matches the original expression.
3. **\( 9x(x - 25) \)**: This expands to \( 9x^2 - 225x \), which is also not equivalent to \( 9x^2 - 25 \).
4. **\( (4.5x + 12.5)(4.5x - 12.5) \)**: This expands to a different expression. It calculates as \( (4.5x)^2 - (12.5)^2 = 20.25x^2 - 156.25 \), which does not match \( 9x^2 - 25 \).
Based on this analysis, the correct answer is:
**\( (3x + 5)(3x - 5) \)**.
Answered by
GPT-4o mini
To verify which expression is equivalent to \( (x - 5)(x - 3) \) using the Distributive Property, we will expand the expression step by step.
Using the Distributive Property, we can expand \( (x - 5)(x - 3) \):
1. **Distribute \( x \):**
- \( x \times x = x^2 \)
- \( x \times (-3) = -3x \)
2. **Distribute \( -5 \):**
- \( -5 \times x = -5x \)
- \( -5 \times (-3) = 15 \)
Now, combine all of these results:
\[
x^2 - 3x - 5x + 15
\]
Next, combine the like terms \( -3x \) and \( -5x \):
\[
x^2 - 8x + 15
\]
So, the expanded form of \( (x - 5)(x - 3) \) is:
\[
x^2 - 8x + 15
\]
Now, let's check the provided responses:
1. **\( x^2 - 8x + 15 \)** - This matches our calculated expansion.
2. **\( 3x^2 - 4x^2 - 3x - 15 \)** - This simplifies to \( -x^2 - 3x - 15 \), which does not match.
3. **\( x^2 - 2x + 15 \)** - This does not match our result.
4. **\( 2x - 2 \)** - This does not match our result as well.
Therefore, the correct answer is:
**\( x^2 - 8x + 15 \)**.
Using the Distributive Property, we can expand \( (x - 5)(x - 3) \):
1. **Distribute \( x \):**
- \( x \times x = x^2 \)
- \( x \times (-3) = -3x \)
2. **Distribute \( -5 \):**
- \( -5 \times x = -5x \)
- \( -5 \times (-3) = 15 \)
Now, combine all of these results:
\[
x^2 - 3x - 5x + 15
\]
Next, combine the like terms \( -3x \) and \( -5x \):
\[
x^2 - 8x + 15
\]
So, the expanded form of \( (x - 5)(x - 3) \) is:
\[
x^2 - 8x + 15
\]
Now, let's check the provided responses:
1. **\( x^2 - 8x + 15 \)** - This matches our calculated expansion.
2. **\( 3x^2 - 4x^2 - 3x - 15 \)** - This simplifies to \( -x^2 - 3x - 15 \), which does not match.
3. **\( x^2 - 2x + 15 \)** - This does not match our result.
4. **\( 2x - 2 \)** - This does not match our result as well.
Therefore, the correct answer is:
**\( x^2 - 8x + 15 \)**.
Answered by
GPT-4o mini
To find the expression equivalent to \( (z + 4)(z - 4) \), we can use the formula for the difference of squares:
\[
(a + b)(a - b) = a^2 - b^2
\]
In this case:
- \( a = z \)
- \( b = 4 \)
Applying the formula:
\[
(z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16
\]
Now, let's evaluate the provided responses:
1. **\( z^2 + 4z - 4z - 16 \)**: This simplifies to \( z^2 - 16 \). (The \( +4z \) and \( -4z \) cancel out.)
2. **\( z(z) + 4z(z) - 4(4) \)**: This expands as \( z^2 + 4z^2 - 16 \). Combining like terms gives \( 5z^2 - 16 \), which is not correct.
3. **\( z^2 - 8z - 16 \)**: This is not correct since it introduces an unwanted \( -8z \) term.
4. **\( 2z^2 - z^2 - 8 \)**: This simplifies to \( z^2 - 8 \), which is also not correct.
Based on the calculations, the equivalent expression for \( (z + 4)(z - 4) \) is:
**\( z^2 - 16 \)**, which corresponds correctly to the simplified form of the first response \( z^2 + 4z - 4z - 16 \) (as all terms confirmed).
\[
(a + b)(a - b) = a^2 - b^2
\]
In this case:
- \( a = z \)
- \( b = 4 \)
Applying the formula:
\[
(z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16
\]
Now, let's evaluate the provided responses:
1. **\( z^2 + 4z - 4z - 16 \)**: This simplifies to \( z^2 - 16 \). (The \( +4z \) and \( -4z \) cancel out.)
2. **\( z(z) + 4z(z) - 4(4) \)**: This expands as \( z^2 + 4z^2 - 16 \). Combining like terms gives \( 5z^2 - 16 \), which is not correct.
3. **\( z^2 - 8z - 16 \)**: This is not correct since it introduces an unwanted \( -8z \) term.
4. **\( 2z^2 - z^2 - 8 \)**: This simplifies to \( z^2 - 8 \), which is also not correct.
Based on the calculations, the equivalent expression for \( (z + 4)(z - 4) \) is:
**\( z^2 - 16 \)**, which corresponds correctly to the simplified form of the first response \( z^2 + 4z - 4z - 16 \) (as all terms confirmed).
Answered by
GPT-4o mini
To find another way to write the expression \( 2(x + 1)(x + 2) \), we will first expand the expression inside the parentheses and then multiply by 2.
1. Begin by expanding \( (x + 1)(x + 2) \) using the distributive property (or FOIL method):
\[
(x + 1)(x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2
\]
2. Now, multiply the entire expression by 2:
\[
2(x^2 + 3x + 2) = 2x^2 + 6x + 4
\]
Now, let's analyze the provided responses:
1. **\( 2x^2 + 5x + 2 \)**: This does not match our result.
2. **\( x^2 + 6x + 2 \)**: This does not match our result.
3. **\( 2x^2 + 3x + 2 \)**: This does not match our result.
4. **\( 2x^2 + 6x + 4 \)**: This matches our result.
Therefore, the correct answer is:
**\( 2x^2 + 6x + 4 \)**.
1. Begin by expanding \( (x + 1)(x + 2) \) using the distributive property (or FOIL method):
\[
(x + 1)(x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2
\]
2. Now, multiply the entire expression by 2:
\[
2(x^2 + 3x + 2) = 2x^2 + 6x + 4
\]
Now, let's analyze the provided responses:
1. **\( 2x^2 + 5x + 2 \)**: This does not match our result.
2. **\( x^2 + 6x + 2 \)**: This does not match our result.
3. **\( 2x^2 + 3x + 2 \)**: This does not match our result.
4. **\( 2x^2 + 6x + 4 \)**: This matches our result.
Therefore, the correct answer is:
**\( 2x^2 + 6x + 4 \)**.
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