Gina is asked to identify a rational expression. Which of the following could she choose?(1 point)

Question

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

Answer 1

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 \( \frac{5.3}{2x - 1} \)).

Answer 2

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

Answer 3

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:

\( \frac{\sqrt{x}}{2x + 2} \).

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

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:

\(-3x - 6\).

Answer 8

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

Answer 9

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.