Asked by Nanika Zoldyck
Gina is asked to identify a rational expression. Which of the following could she choose?(1 point)
Responses
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
x32+52x9
Start Fraction x superscript Start Fraction 3 over 2 End Fraction baseline plus 5 over 2 x superscript 9 baseline End Fraction
5.32x−1
Start Fraction 5.3 over 2 x minus 1 End Fraction
8x2+70
Responses
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
x32+52x9
Start Fraction x superscript Start Fraction 3 over 2 End Fraction baseline plus 5 over 2 x superscript 9 baseline End Fraction
5.32x−1
Start Fraction 5.3 over 2 x minus 1 End Fraction
8x2+70
Answers
Answered by
Nanika Zoldyck
Which of the following is a rational expression whose domain has a restriction of x≠−1 ?(1 point)
Responses
x+1x3
Start Fraction x plus 1 over x cubed End Fraction
6x3+60x
Start Fraction 6 x cubed plus 60 over x End Fraction
x√2x+2
Start Fraction Start Root x End Root over 2 x plus 2 End Fraction
x2+6x+56x+6
Responses
x+1x3
Start Fraction x plus 1 over x cubed End Fraction
6x3+60x
Start Fraction 6 x cubed plus 60 over x End Fraction
x√2x+2
Start Fraction Start Root x End Root over 2 x plus 2 End Fraction
x2+6x+56x+6
Answered by
Nanika Zoldyck
Which of the following is a simplified, equivalent form of the rational expression 5x3−8x2+34x2 ?(1 point)
Responses
5x34−2x2+34x2
Start Fraction 5 x cubed over 4 End Fraction minus 2 x squared plus Start Fraction 3 over 4 x squared End Fraction
5x4−2+34x2
Start Fraction 5 x over 4 End Fraction minus 2 plus Start Fraction 3 over 4 x squared End Fraction
5x54−2x4+34x2
Start Fraction 5 x superscript 5 baseline over 4 End Fraction minus 2 x superscript 4 baseline plus Start Fraction 3 over 4 x squared End Fraction
5x4+34x2
Responses
5x34−2x2+34x2
Start Fraction 5 x cubed over 4 End Fraction minus 2 x squared plus Start Fraction 3 over 4 x squared End Fraction
5x4−2+34x2
Start Fraction 5 x over 4 End Fraction minus 2 plus Start Fraction 3 over 4 x squared End Fraction
5x54−2x4+34x2
Start Fraction 5 x superscript 5 baseline over 4 End Fraction minus 2 x superscript 4 baseline plus Start Fraction 3 over 4 x squared End Fraction
5x4+34x2
Answered by
Nanika Zoldyck
Melany was asked to create an equivalent form of the rational expression −3x2−15x−18x+3 . Which of the following is a correct equivalent form?(1 point)
Responses
This rational expression does not have any equivalent forms.
This rational expression does not have any equivalent forms.
−3x−24−114x+3
negative 3 x minus 24 minus Start Fraction 114 over x plus 3 End Fraction
−3x−114
negative 3 x minus 114
−3x−6
Responses
This rational expression does not have any equivalent forms.
This rational expression does not have any equivalent forms.
−3x−24−114x+3
negative 3 x minus 24 minus Start Fraction 114 over x plus 3 End Fraction
−3x−114
negative 3 x minus 114
−3x−6
Answered by
Nanika Zoldyck
Which of the following expressions simplifies to an equivalent simplified form of 2x3−2xx+1 ?(1 point)
Responses
x2−12x2+2x
Start Fraction x squared minus 1 over 2 x squared plus 2 x End Fraction
2x2+6xx2+2x−3
Start Fraction 2 x squared plus 6 x over x squared plus 2 x minus 3 End Fraction
2x3−2xx−1
Start Fraction 2 x cubed minus 2 x over x minus 1 End Fraction
2x3+2x2−4xx+2
Responses
x2−12x2+2x
Start Fraction x squared minus 1 over 2 x squared plus 2 x End Fraction
2x2+6xx2+2x−3
Start Fraction 2 x squared plus 6 x over x squared plus 2 x minus 3 End Fraction
2x3−2xx−1
Start Fraction 2 x cubed minus 2 x over x minus 1 End Fraction
2x3+2x2−4xx+2
Answered by
GPT-4o mini
A rational expression is a fraction in which both the numerator and the denominator are polynomials.
Let's examine each option:
1. **9x−48+x√**: This expression is not presented correctly, but based on my understanding, if it were meant to be a simplified rational expression, it doesn’t appear to be in the form of a polynomial over a polynomial due to the presence of the square root.
2. **x32+52x9**: This doesn't appear to be in standard form, but if it were meant to be \( \frac{x^{\frac{3}{2}} + \frac{5}{2}x^9}{something} \), then we'd need the denominator to determine if it's a rational expression. Without a denominator shown, we can't confirm it's a rational expression.
3. **5.32x−1**: This appears to be in the form of a fraction, likely \( \frac{5.3}{2x - 1} \). Here, both the numerator (5.3, which can be treated as a constant polynomial) and the denominator (2x - 1, which is a polynomial) fit the definition of a rational expression.
4. **8x2+70**: This expression is simply a polynomial (without a denominator), so it is not a rational expression by itself.
Based on these analyses, the most likely choice that represents a rational expression is:
**5.32x−1** (interpreted as \( \frac{5.3}{2x - 1} \)).
Let's examine each option:
1. **9x−48+x√**: This expression is not presented correctly, but based on my understanding, if it were meant to be a simplified rational expression, it doesn’t appear to be in the form of a polynomial over a polynomial due to the presence of the square root.
2. **x32+52x9**: This doesn't appear to be in standard form, but if it were meant to be \( \frac{x^{\frac{3}{2}} + \frac{5}{2}x^9}{something} \), then we'd need the denominator to determine if it's a rational expression. Without a denominator shown, we can't confirm it's a rational expression.
3. **5.32x−1**: This appears to be in the form of a fraction, likely \( \frac{5.3}{2x - 1} \). Here, both the numerator (5.3, which can be treated as a constant polynomial) and the denominator (2x - 1, which is a polynomial) fit the definition of a rational expression.
4. **8x2+70**: This expression is simply a polynomial (without a denominator), so it is not a rational expression by itself.
Based on these analyses, the most likely choice that represents a rational expression is:
**5.32x−1** (interpreted as \( \frac{5.3}{2x - 1} \)).
Answered by
GPT-4o mini
To determine which of the given options is a rational expression with a domain restriction of \( x \neq -1 \), we need to look for a rational expression where the denominator can equal zero when \( x = -1 \).
Let's analyze each option:
1. **\( \frac{x + 1}{x^3} \)**: The denominator is \( x^3 \), which equals zero when \( x = 0 \). The numerator \( x + 1 \) equals zero when \( x = -1 \), but this does not restrict the domain because it does not make the entire expression undefined. The restriction here is \( x \neq 0 \).
2. **\( \frac{6x^3 + 60}{x} \)**: The denominator is \( x \), which equals zero when \( x = 0 \), leading to the restriction \( x \neq 0 \). There is no restriction at \( x = -1 \) here.
3. **\( \frac{\sqrt{x}}{2x + 2} \)**: The denominator \( 2x + 2 \) equals zero when \( x = -1 \) (since \( 2(-1) + 2 = 0 \)). Therefore, there is a restriction at \( x = -1 \). We also have \( \sqrt{x} \) which restricts \( x \) to be non-negative (i.e., \( x \geq 0 \)), but the main restriction from the denominator is \( x \neq -1 \).
4. **\( \frac{x^2 + 6x + 5}{x + 6} \)**: The denominator \( x + 6 \) equals zero when \( x = -6 \), and does not create a restriction at \( x = -1 \).
Based on the analysis, the expression that clearly has a domain restriction of \( x \neq -1 \) is:
**\( \frac{\sqrt{x}}{2x + 2} \)**.
Let's analyze each option:
1. **\( \frac{x + 1}{x^3} \)**: The denominator is \( x^3 \), which equals zero when \( x = 0 \). The numerator \( x + 1 \) equals zero when \( x = -1 \), but this does not restrict the domain because it does not make the entire expression undefined. The restriction here is \( x \neq 0 \).
2. **\( \frac{6x^3 + 60}{x} \)**: The denominator is \( x \), which equals zero when \( x = 0 \), leading to the restriction \( x \neq 0 \). There is no restriction at \( x = -1 \) here.
3. **\( \frac{\sqrt{x}}{2x + 2} \)**: The denominator \( 2x + 2 \) equals zero when \( x = -1 \) (since \( 2(-1) + 2 = 0 \)). Therefore, there is a restriction at \( x = -1 \). We also have \( \sqrt{x} \) which restricts \( x \) to be non-negative (i.e., \( x \geq 0 \)), but the main restriction from the denominator is \( x \neq -1 \).
4. **\( \frac{x^2 + 6x + 5}{x + 6} \)**: The denominator \( x + 6 \) equals zero when \( x = -6 \), and does not create a restriction at \( x = -1 \).
Based on the analysis, the expression that clearly has a domain restriction of \( x \neq -1 \) is:
**\( \frac{\sqrt{x}}{2x + 2} \)**.
Answered by
GPT-4o mini
To determine which option is a simplified, equivalent form of the expression \( 5x^3 - 8x^2 + 34x^2 \), we first need to simplify the given expression itself.
Let's combine like terms in the original expression:
1. Start with the expression:
\[
5x^3 - 8x^2 + 34x^2
\]
2. Combine the \( x^2 \) terms:
\[
-8x^2 + 34x^2 = (34 - 8)x^2 = 26x^2
\]
So, the simplified form of the expression is:
\[
5x^3 + 26x^2
\]
Now, let's analyze each of the provided response options to see if any match this simplified form:
1. **\( \frac{5x^3}{4} - 2x^2 + \frac{3}{4}x^2 \)**:
Simplifying this:
\[
-2x^2 + \frac{3}{4}x^2 = (-2 + \frac{3}{4})x^2 = (-\frac{8}{4} + \frac{3}{4})x^2 = -\frac{5}{4}x^2
\]
This does not match the desired expression.
2. **\( \frac{5x}{4} - 2 + \frac{3}{4}x^2 \)**:
This expression has different powers of \( x \) and does not include \( 5x^3 \), so this is not a match.
3. **\( \frac{5x^5}{4} - 2x^4 + \frac{3}{4}x^2 \)**:
Again, this expression has higher powers of \( x \) and does not match.
4. **\( 5x^4 + 34x^2 \)**:
This expression again has the wrong degree for the \( x^3 \) term.
After analyzing all the provided options, none of them represent a simplified, equivalent form of the expression \( 5x^3 + 26x^2 \) correctly.
Given this, it seems like none of the provided options are correct. Please verify the original expression or options, as they may need to be revised.
Let's combine like terms in the original expression:
1. Start with the expression:
\[
5x^3 - 8x^2 + 34x^2
\]
2. Combine the \( x^2 \) terms:
\[
-8x^2 + 34x^2 = (34 - 8)x^2 = 26x^2
\]
So, the simplified form of the expression is:
\[
5x^3 + 26x^2
\]
Now, let's analyze each of the provided response options to see if any match this simplified form:
1. **\( \frac{5x^3}{4} - 2x^2 + \frac{3}{4}x^2 \)**:
Simplifying this:
\[
-2x^2 + \frac{3}{4}x^2 = (-2 + \frac{3}{4})x^2 = (-\frac{8}{4} + \frac{3}{4})x^2 = -\frac{5}{4}x^2
\]
This does not match the desired expression.
2. **\( \frac{5x}{4} - 2 + \frac{3}{4}x^2 \)**:
This expression has different powers of \( x \) and does not include \( 5x^3 \), so this is not a match.
3. **\( \frac{5x^5}{4} - 2x^4 + \frac{3}{4}x^2 \)**:
Again, this expression has higher powers of \( x \) and does not match.
4. **\( 5x^4 + 34x^2 \)**:
This expression again has the wrong degree for the \( x^3 \) term.
After analyzing all the provided options, none of them represent a simplified, equivalent form of the expression \( 5x^3 + 26x^2 \) correctly.
Given this, it seems like none of the provided options are correct. Please verify the original expression or options, as they may need to be revised.
Answered by
GPT-4o mini
To find an equivalent form of the rational expression \(-\frac{3x^2 - 15x - 18}{x + 3}\), we first need to simplify the expression.
### Step 1: Factor the Numerator
The numerator is \(-3x^2 - 15x - 18\). We can factor it.
1. Factor out \(-3\) from the numerator:
\[
-3(x^2 + 5x + 6)
\]
2. Now, factor the quadratic \(x^2 + 5x + 6\):
The expression factors to:
\[
(x + 2)(x + 3)
\]
So, the entire expression becomes:
\[
-\frac{3(x + 2)(x + 3)}{x + 3}
\]
### Step 2: Simplify
Upon simplification, as long as \(x \neq -3\) (to avoid division by zero), we can cancel \(x + 3\) from the numerator and denominator:
\[
-3(x + 2)
\]
### Resulting Expression
Thus, the simplified form of the expression is:
\[
-3(x + 2) = -3x - 6
\]
### Analyzing the Options
Now let’s analyze the response options:
1. **This rational expression does not have any equivalent forms.**
This is incorrect because we found an equivalent form.
2. **\(-3x - 24 - \frac{114}{x + 3}\)**:
This cannot be simplified to our result.
3. **\(-3x - 114\)**:
This does not match our resulting expression.
4. **\(-3x - 6\)**:
This matches the simplified form we derived.
### Conclusion
Thus, the correct equivalent form of the rational expression is:
**\(-3x - 6\)**.
### Step 1: Factor the Numerator
The numerator is \(-3x^2 - 15x - 18\). We can factor it.
1. Factor out \(-3\) from the numerator:
\[
-3(x^2 + 5x + 6)
\]
2. Now, factor the quadratic \(x^2 + 5x + 6\):
The expression factors to:
\[
(x + 2)(x + 3)
\]
So, the entire expression becomes:
\[
-\frac{3(x + 2)(x + 3)}{x + 3}
\]
### Step 2: Simplify
Upon simplification, as long as \(x \neq -3\) (to avoid division by zero), we can cancel \(x + 3\) from the numerator and denominator:
\[
-3(x + 2)
\]
### Resulting Expression
Thus, the simplified form of the expression is:
\[
-3(x + 2) = -3x - 6
\]
### Analyzing the Options
Now let’s analyze the response options:
1. **This rational expression does not have any equivalent forms.**
This is incorrect because we found an equivalent form.
2. **\(-3x - 24 - \frac{114}{x + 3}\)**:
This cannot be simplified to our result.
3. **\(-3x - 114\)**:
This does not match our resulting expression.
4. **\(-3x - 6\)**:
This matches the simplified form we derived.
### Conclusion
Thus, the correct equivalent form of the rational expression is:
**\(-3x - 6\)**.
Answered by
GPT-4o mini
To determine which of the provided expressions simplifies to an equivalent form of \( \frac{2x^3 - 2x}{x + 1} \), we will start by simplifying this expression.
### Step 1: Simplify the Expression
First, factor the numerator \( 2x^3 - 2x \):
1. Factor out the common factor \( 2x \):
\[
2x(x^2 - 1)
\]
2. Notice that \( x^2 - 1 \) is a difference of squares, so we can factor it further:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Therefore, the expression can be rewritten as:
\[
\frac{2x(x - 1)(x + 1)}{x + 1}
\]
### Step 2: Cancel Common Factors
As long as \( x \neq -1 \) (to avoid division by zero), we can cancel \( x + 1 \) from the numerator and denominator:
\[
2x(x - 1)
\]
### Resulting Expression
Thus, the simplified form is:
\[
2x^2 - 2x
\]
### Step 3: Verify the Given Options
Now let's check the provided responses to see which one matches \( 2x^2 - 2x \).
1. **\( \frac{x^2 - 1}{2x^2 + 2x} \)**:
This expression is not equivalent.
2. **\( \frac{2x^2 + 6x}{x^2 + 2x - 3} \)**:
Factor the numerator and the denominator:
- \(2x^2 + 6x = 2x(x + 3)\)
- \(x^2 + 2x - 3 = (x + 3)(x - 1)\)
Thus,
\[
\frac{2x(x + 3)}{(x + 3)(x - 1)} = \frac{2x}{x - 1} \quad \text{(for } x \neq -3 \text{)}
\]
This does not match.
3. **\( \frac{2x^3 - 2x}{x - 1} \)**:
This does not match our resulting simplified expression since the denominator should be \( x + 1 \).
4. **\( 2x^3 + 2x^2 - 4x + 2 \)**:
This expression can be simplified as follows:
\[
2x^3 + 2x^2 - 4x + 2 = 2(x^3 + x^2 - 2x + 1)
\]
However, this does not factor or simplify to \( 2x^2 - 2x \).
### Conclusion
None of the provided options match \( 2x^2 - 2x \). It appears that there is no equivalent expression in the responses given. Please verify the original expression or provided options if any corrections are needed.
### Step 1: Simplify the Expression
First, factor the numerator \( 2x^3 - 2x \):
1. Factor out the common factor \( 2x \):
\[
2x(x^2 - 1)
\]
2. Notice that \( x^2 - 1 \) is a difference of squares, so we can factor it further:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Therefore, the expression can be rewritten as:
\[
\frac{2x(x - 1)(x + 1)}{x + 1}
\]
### Step 2: Cancel Common Factors
As long as \( x \neq -1 \) (to avoid division by zero), we can cancel \( x + 1 \) from the numerator and denominator:
\[
2x(x - 1)
\]
### Resulting Expression
Thus, the simplified form is:
\[
2x^2 - 2x
\]
### Step 3: Verify the Given Options
Now let's check the provided responses to see which one matches \( 2x^2 - 2x \).
1. **\( \frac{x^2 - 1}{2x^2 + 2x} \)**:
This expression is not equivalent.
2. **\( \frac{2x^2 + 6x}{x^2 + 2x - 3} \)**:
Factor the numerator and the denominator:
- \(2x^2 + 6x = 2x(x + 3)\)
- \(x^2 + 2x - 3 = (x + 3)(x - 1)\)
Thus,
\[
\frac{2x(x + 3)}{(x + 3)(x - 1)} = \frac{2x}{x - 1} \quad \text{(for } x \neq -3 \text{)}
\]
This does not match.
3. **\( \frac{2x^3 - 2x}{x - 1} \)**:
This does not match our resulting simplified expression since the denominator should be \( x + 1 \).
4. **\( 2x^3 + 2x^2 - 4x + 2 \)**:
This expression can be simplified as follows:
\[
2x^3 + 2x^2 - 4x + 2 = 2(x^3 + x^2 - 2x + 1)
\]
However, this does not factor or simplify to \( 2x^2 - 2x \).
### Conclusion
None of the provided options match \( 2x^2 - 2x \). It appears that there is no equivalent expression in the responses given. Please verify the original expression or provided options if any corrections are needed.
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