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
x+3x+1
Start Fraction x plus 3 over x plus 1 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
x2−33(x−3)(x+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
x+3x+1
Start Fraction x plus 3 over x plus 1 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
x2−33(x−3)(x+1)
Answers
GPT-4o mini
To subtract the rational expressions given, we first need to clarify and simplify the expression:
\[
\frac{x^2 + 4x - 21}{x^2 - 2x - 3} - (4x + 1)
\]
We'll start by factoring the polynomial in the denominator and numerator if possible.
1. **Factor the denominator**:
- \(x^2 - 2x - 3\) can be factored into \((x - 3)(x + 1)\).
2. **Factor the numerator of the rational expression**:
- \(x^2 + 4x - 21\) can be factored into \((x + 7)(x - 3)\).
Now rewrite the original expression:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - (4x + 1)
\]
3. **Combine the terms**:
To perform the subtraction, we need to have a common denominator. The common denominator is \((x - 3)(x + 1)\).
Now let's rewrite \(4x + 1\) with this denominator:
\[
4x + 1 = \frac{(4x + 1)(x - 3)(x + 1)}{(x - 3)(x + 1)}
\]
Now expand \((4x + 1)(x + 1)\):
\[
(4x + 1)(x + 1) = 4x^2 + 5x + 1
\]
Now combine:
\[
4x - 3 = 4x^2 + 5x + 1
\]
Therefore, subtract:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - \frac{4x^2 + 5x + 1}{(x - 3)(x + 1)} = \frac{(x + 7)(x - 3) - (4x^2 + 5x + 1)}{(x - 3)(x + 1)}
\]
4. **Simplify the numerator**:
Expanding \((x + 7)(x - 3)\) gives:
\[
x^2 - 3x + 7x - 21 = x^2 + 4x - 21
\]
Combine:
\[
x^2 + 4x - 21 - 4x^2 - 5x - 1 = -3x^2 - 22 + x = -3x^2 - 18
\]
Thus we have:
\[
\frac{-3x^2 - 18}{(x - 3)(x + 1)}
\]
5. **Factor out the numerator**:
You can factor out \(-3\):
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
The numerator does not factor further, so we have:
Final expression:
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
The simplified difference is
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
Comparing with your options, the appropriate response matches closely with one of the choices but does not explicitly match any of the provided responses provided in your list.
**Final Answer**:
Try finding and confirming the options provided, or answer according to your context from the expressions.
\[
\frac{x^2 + 4x - 21}{x^2 - 2x - 3} - (4x + 1)
\]
We'll start by factoring the polynomial in the denominator and numerator if possible.
1. **Factor the denominator**:
- \(x^2 - 2x - 3\) can be factored into \((x - 3)(x + 1)\).
2. **Factor the numerator of the rational expression**:
- \(x^2 + 4x - 21\) can be factored into \((x + 7)(x - 3)\).
Now rewrite the original expression:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - (4x + 1)
\]
3. **Combine the terms**:
To perform the subtraction, we need to have a common denominator. The common denominator is \((x - 3)(x + 1)\).
Now let's rewrite \(4x + 1\) with this denominator:
\[
4x + 1 = \frac{(4x + 1)(x - 3)(x + 1)}{(x - 3)(x + 1)}
\]
Now expand \((4x + 1)(x + 1)\):
\[
(4x + 1)(x + 1) = 4x^2 + 5x + 1
\]
Now combine:
\[
4x - 3 = 4x^2 + 5x + 1
\]
Therefore, subtract:
\[
\frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - \frac{4x^2 + 5x + 1}{(x - 3)(x + 1)} = \frac{(x + 7)(x - 3) - (4x^2 + 5x + 1)}{(x - 3)(x + 1)}
\]
4. **Simplify the numerator**:
Expanding \((x + 7)(x - 3)\) gives:
\[
x^2 - 3x + 7x - 21 = x^2 + 4x - 21
\]
Combine:
\[
x^2 + 4x - 21 - 4x^2 - 5x - 1 = -3x^2 - 22 + x = -3x^2 - 18
\]
Thus we have:
\[
\frac{-3x^2 - 18}{(x - 3)(x + 1)}
\]
5. **Factor out the numerator**:
You can factor out \(-3\):
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
The numerator does not factor further, so we have:
Final expression:
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
The simplified difference is
\[
\frac{-3(x^2 + 6)}{(x - 3)(x + 1)}
\]
Comparing with your options, the appropriate response matches closely with one of the choices but does not explicitly match any of the provided responses provided in your list.
**Final Answer**:
Try finding and confirming the options provided, or answer according to your context from the expressions.