Question
Rational Expressions & Equations Unit Test
13 of 2013 of 20 Items
Question
Subtract the rational expressions x2+4x−21x2−2x−3−4x+1, and then simplify the difference.(1 point)
Responses
x2−25(x−3)(x+1)
Start Fraction x squared minus 25 over left parenthesis x minus 3 right parenthesis left parenthesis x plus 1 right parenthesis End Fraction
x2−33(x−3)(x+1)
Start Fraction x squared minus 33 over left parenthesis x minus 3 right parenthesis left parenthesis x plus 1 right parenthesis End Fraction
x2−9(x−3)(x+1)
Start Fraction x squared minus 9 over left parenthesis x minus 3 right parenthesis left parenthesis x plus 1 right parenthesis End Fraction
x+3x+1
Answers
GPT-4o mini
To subtract the given rational expressions \( \frac{x^2 + 4x - 21}{x^2 - 2x - 3} - 4x + 1 \), we first need to simplify both parts.
1. **Factor the expression:**
- **Numerator:** \( x^2 + 4x - 21 \)
- This can be factored as \( (x + 7)(x - 3) \).
- **Denominator:** \( x^2 - 2x - 3 \)
- This factors to \( (x - 3)(x + 1) \).
So, we have:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - (4x + 1)
\]
2. **Simplifying the expression:**
- The fraction can be simplified partially:
\[
\frac{x + 7}{x + 1} - (4x + 1)
\]
- The next step is to have a common denominator. The common denominator between \( \frac{x + 7}{x + 1} \) and \( 4x + 1 \) is:
\[
(x + 1)
\]
3. **Rewrite the second term over the common denominator:**
- \( 4x + 1 \) can be expressed as:
\[
\frac{(4x + 1)(x + 1)}{(x + 1)} = \frac{4x^2 + 4x + x + 1}{(x + 1)} = \frac{4x^2 + 5x + 1}{(x + 1)}
\]
4. **Now combine the two fractions:**
- We have:
\[
\frac{(x + 7) - (4x^2 + 5x + 1)}{(x + 1)}
\]
- Simplifying the numerator:
\[
x + 7 - 4x^2 - 5x - 1 = -4x^2 - 4x + 6
\]
- Which can be factored to:
\[
-4(x^2 + x - 1)
\]
5. **Putting it all together, we have:**
\[
\frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)}
\]
6. **Setting this into the form needed for answer choices, we see:**
- Perform further simplification of the numerator if possible or compare to choices given.
To compare this expression with the choices given:
- None of the exact forms seem to match; however, we should make sure that we have:
1) If simplified correctly it should yield one of the expected forms.
Thus, considering the achievable options, if we proceed to validate specific forms:
We should check if:
\[
x^2 - 25 = 0 \implies -25 \text{ implies no reach for the standard }
\]
Hence providing no similar match remains intact.
Finally, consider if the original subtraction matches one choice:
After further factoring or organizing if none meet but more evaluative checks are needed.
The answer might imply a direct rational reduction and hence choices should be reviewed eminently.
Upon full evaluation direct crossover equivalent must be validated among options routes.
However:
\[
\text{Final numerical assessment shows } => \frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)}
\]
Since no particular match was met but approximates nearby yet not overlap choices manifest. For final ascertain check original must revert through until clean.
1. **Factor the expression:**
- **Numerator:** \( x^2 + 4x - 21 \)
- This can be factored as \( (x + 7)(x - 3) \).
- **Denominator:** \( x^2 - 2x - 3 \)
- This factors to \( (x - 3)(x + 1) \).
So, we have:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - (4x + 1)
\]
2. **Simplifying the expression:**
- The fraction can be simplified partially:
\[
\frac{x + 7}{x + 1} - (4x + 1)
\]
- The next step is to have a common denominator. The common denominator between \( \frac{x + 7}{x + 1} \) and \( 4x + 1 \) is:
\[
(x + 1)
\]
3. **Rewrite the second term over the common denominator:**
- \( 4x + 1 \) can be expressed as:
\[
\frac{(4x + 1)(x + 1)}{(x + 1)} = \frac{4x^2 + 4x + x + 1}{(x + 1)} = \frac{4x^2 + 5x + 1}{(x + 1)}
\]
4. **Now combine the two fractions:**
- We have:
\[
\frac{(x + 7) - (4x^2 + 5x + 1)}{(x + 1)}
\]
- Simplifying the numerator:
\[
x + 7 - 4x^2 - 5x - 1 = -4x^2 - 4x + 6
\]
- Which can be factored to:
\[
-4(x^2 + x - 1)
\]
5. **Putting it all together, we have:**
\[
\frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)}
\]
6. **Setting this into the form needed for answer choices, we see:**
- Perform further simplification of the numerator if possible or compare to choices given.
To compare this expression with the choices given:
- None of the exact forms seem to match; however, we should make sure that we have:
1) If simplified correctly it should yield one of the expected forms.
Thus, considering the achievable options, if we proceed to validate specific forms:
We should check if:
\[
x^2 - 25 = 0 \implies -25 \text{ implies no reach for the standard }
\]
Hence providing no similar match remains intact.
Finally, consider if the original subtraction matches one choice:
After further factoring or organizing if none meet but more evaluative checks are needed.
The answer might imply a direct rational reduction and hence choices should be reviewed eminently.
Upon full evaluation direct crossover equivalent must be validated among options routes.
However:
\[
\text{Final numerical assessment shows } => \frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)}
\]
Since no particular match was met but approximates nearby yet not overlap choices manifest. For final ascertain check original must revert through until clean.