Question
Add or subtract. Simplify if possible. p2−11p+28p2−4p−21−6p+3 (1 point) Responses p−4p+3 the fraction with numerator p minus 4 and denominator p plus 3 p−10p+3 the fraction with numerator p minus 10 and denominator p plus 3 p−10 p minus 10 p2−11p+22p2−4p−21
Answers
GPT-4o mini
To simplify the expression
\[
\frac{p^2 - 11p + 28}{p^2 - 4p - 21} - 6p + 3,
\]
let's start by factoring the polynomials in the numerator and the denominator.
1. **Factoring \(p^2 - 11p + 28\)**:
We look for two numbers that multiply to \(28\) and add to \(-11\). These numbers are \(-4\) and \(-7\). So we can factor it as:
\[
p^2 - 11p + 28 = (p - 4)(p - 7).
\]
2. **Factoring \(p^2 - 4p - 21\)**:
We need two numbers that multiply to \(-21\) and add to \(-4\). These numbers are \(-7\) and \(3\). Thus, it can be factored as:
\[
p^2 - 4p - 21 = (p - 7)(p + 3).
\]
Now, substituting these factored forms back into the expression, we get:
\[
\frac{(p - 4)(p - 7)}{(p - 7)(p + 3)} - 6p + 3.
\]
3. **Simplifying the Fraction**:
We can cancel the common factor of \(p - 7\) from the numerator and denominator, assuming \(p \neq 7\):
\[
\frac{p - 4}{p + 3} - 6p + 3.
\]
4. **Combining Terms**:
Next, we want to combine \(\frac{p - 4}{p + 3}\) with \(-6p + 3\). To do this, we express \(-6p + 3\) with a common denominator:
\[
-6p + 3 = \frac{-6p(p + 3) + 3(p + 3)}{p + 3} = \frac{-6p^2 - 18p + 3p + 9}{p + 3} = \frac{-6p^2 - 15p + 9}{p + 3}.
\]
5. **Combining the Two Fractions**:
Now we have:
\[
\frac{p - 4}{p + 3} + \frac{-6p^2 - 15p + 9}{p + 3} = \frac{p - 4 - 6p^2 - 15p + 9}{p + 3}.
\]
This simplifies to:
\[
\frac{-6p^2 - 14p + 5}{p + 3}.
\]
Thus, our final simplified result is:
\[
\frac{-6p^2 - 14p + 5}{p + 3}.
\]
So, the final answer could be none of the options provided. However, if we factor out -1 from the numerator, we could write it as:
\[
-\frac{6p^2 + 14p - 5}{p + 3}.
\]
If this does not match any choices, please provide more context or double-check the options again.
\[
\frac{p^2 - 11p + 28}{p^2 - 4p - 21} - 6p + 3,
\]
let's start by factoring the polynomials in the numerator and the denominator.
1. **Factoring \(p^2 - 11p + 28\)**:
We look for two numbers that multiply to \(28\) and add to \(-11\). These numbers are \(-4\) and \(-7\). So we can factor it as:
\[
p^2 - 11p + 28 = (p - 4)(p - 7).
\]
2. **Factoring \(p^2 - 4p - 21\)**:
We need two numbers that multiply to \(-21\) and add to \(-4\). These numbers are \(-7\) and \(3\). Thus, it can be factored as:
\[
p^2 - 4p - 21 = (p - 7)(p + 3).
\]
Now, substituting these factored forms back into the expression, we get:
\[
\frac{(p - 4)(p - 7)}{(p - 7)(p + 3)} - 6p + 3.
\]
3. **Simplifying the Fraction**:
We can cancel the common factor of \(p - 7\) from the numerator and denominator, assuming \(p \neq 7\):
\[
\frac{p - 4}{p + 3} - 6p + 3.
\]
4. **Combining Terms**:
Next, we want to combine \(\frac{p - 4}{p + 3}\) with \(-6p + 3\). To do this, we express \(-6p + 3\) with a common denominator:
\[
-6p + 3 = \frac{-6p(p + 3) + 3(p + 3)}{p + 3} = \frac{-6p^2 - 18p + 3p + 9}{p + 3} = \frac{-6p^2 - 15p + 9}{p + 3}.
\]
5. **Combining the Two Fractions**:
Now we have:
\[
\frac{p - 4}{p + 3} + \frac{-6p^2 - 15p + 9}{p + 3} = \frac{p - 4 - 6p^2 - 15p + 9}{p + 3}.
\]
This simplifies to:
\[
\frac{-6p^2 - 14p + 5}{p + 3}.
\]
Thus, our final simplified result is:
\[
\frac{-6p^2 - 14p + 5}{p + 3}.
\]
So, the final answer could be none of the options provided. However, if we factor out -1 from the numerator, we could write it as:
\[
-\frac{6p^2 + 14p - 5}{p + 3}.
\]
If this does not match any choices, please provide more context or double-check the options again.