To determine which operation results in the simplified expression [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], let's break down the problem step-by-step.
We are given the polynomials:
[tex]\[ P = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
First, let's simplify [tex]\( Q \)[/tex]:
[tex]\[
Q = (x^3 + 2x^2 + 3)(x^2 - 2)
\][/tex]
Multiplying these polynomials out:
[tex]\[
Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2)
\][/tex]
Distribute each term:
[tex]\[
Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6
\][/tex]
Combine like terms:
[tex]\[
Q = x^5 + 2x^4 - 2x^3 - x^2 - 6
\][/tex]
Now we look at which operation between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] yields [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex]. The possible operations are:
1. [tex]\( PQ \)[/tex]
2. [tex]\( P - Q \)[/tex]
3. [tex]\( Q - P \)[/tex]
4. [tex]\( P + Q \)[/tex]
Given the desired simplified expression, we compare the results of each operation:
1. [tex]\( PQ \)[/tex]:
- Multiplying [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] yields a polynomial much more complex than the target polynomial and of a higher degree (likely degree 9). Therefore, [tex]\( PQ \)[/tex] is not correct.
2. [tex]\( P - Q \)[/tex]:
- Subtracting [tex]\( Q \)[/tex] from [tex]\( P \)[/tex]:
[tex]\[
P - Q = \left(x^4 + 3x^3 + 2x^2 - x + 2\right) - \left(x^5 + 2x^4 - 2x^3 - x^2 - 6\right)
\][/tex]
- Simplify:
[tex]\[
P - Q = -x^5 - x^4 + 5x^3 + 3x^2 - x + 8
\][/tex]
- This transformation gives:
[tex]\[
P - Q = x^5 + x^4 - 5x^3 - 3x^2 + x - 8
\][/tex]
Following these steps, the simplified polynomial [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex] matches the expression found by computing [tex]\( Q - P \)[/tex]. Therefore, the correct operation is:
[tex]\[ \boxed{Q - P} \][/tex]
So, the correct answer is [tex]\( \text{C. } Q - P \)[/tex].
