Answer :
To determine the operation that results in the simplified expression [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], we need to start by simplifying the polynomial [tex]\( Q \)[/tex] and then investigating the given options step-by-step.
Given polynomials:
[tex]\[ P = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
First step: Simplify [tex]\( Q \)[/tex]
Expand the product in [tex]\( Q \)[/tex]:
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[ Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2) \][/tex]
Calculate each term:
[tex]\[ x^3(x^2 - 2) = x^5 - 2x^3 \][/tex]
[tex]\[ 2x^2(x^2 - 2) = 2x^4 - 4x^2 \][/tex]
[tex]\[ 3(x^2 - 2) = 3x^2 - 6 \][/tex]
Combine these:
[tex]\[ Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \][/tex]
[tex]\[ Q = x^5 + 2x^4 - 2x^3 - x^2 - 6 \][/tex]
Now, let's consider the different operations given in the options and see which one simplifies to [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex].
### Option A: [tex]\( P + Q \)[/tex]
Calculate [tex]\( P + Q \)[/tex]:
[tex]\[ P + Q = (x^4 + 3x^3 + 2x^2 - x + 2) + (x^5 + 2x^4 - 2x^3 - x^2 - 6) \][/tex]
Combine like terms:
[tex]\[ P + Q = x^5 + (x^4 + 2x^4) + (3x^3 - 2x^3) + (2x^2 - x^2) - x + (2 - 6) \][/tex]
[tex]\[ P + Q = x^5 + 3x^4 + x^3 + x^2 - x - 4 \][/tex]
This is different from [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], so option A is incorrect.
### Option B: [tex]\( Q - P \)[/tex]
Calculate [tex]\( Q - P \)[/tex]:
[tex]\[ Q - P = (x^5 + 2x^4 - 2x^3 - x^2 - 6) - (x^4 + 3x^3 + 2x^2 - x + 2) \][/tex]
Combine like terms:
[tex]\[ Q - P = x^5 + (2x^4 - x^4) + (-2x^3 - 3x^3) + (-x^2 - 2x^2) + (-x) + (-6 - 2) \][/tex]
[tex]\[ Q - P = x^5 + x^4 - 5x^3 - 3x^2 - x - 8 \][/tex]
This closely matches [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex] except the sign of 'x', so option B is incorrect.
### Option C: [tex]\( P - Q \)[/tex]
Calculate [tex]\( P - Q \)[/tex]:
[tex]\[ P - Q = (x^4 + 3x^3 + 2x^2 - x + 2) - (x^5 + 2x^4 - 2x^3 - x^2 - 6) \][/tex]
Combine like terms:
[tex]\[ P - Q = -x^5 + (x^4 - 2x^4) + (3x^3 + 2x^3) + (2x^2 - (-x^2)) - x - (-6 + 2) \][/tex]
[tex]\[ P - Q = -x^5 - x^4 + 5x^3 + 3x^2 - x + 8 \][/tex]
This does not match [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], so option C is incorrect.
### Option D: [tex]\( PQ \)[/tex]
Lastly, multiplying [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
The degree of the resultant polynomial from [tex]\( PQ \)[/tex] would be [tex]\( 4 + 5 = 9 \)[/tex](multiplying polynomials). So this cannot be a degree 5 polynomial:
Hence, option D is incorrect.
### Conclusion:
Through elimination and verification of the polynomial equation:
A. Incorrect
B. Incorrect
C. Incorrect
D. Incorrect
There might be some consistent steps which we need to tighten our results by recalculation but from polynomial operations checks verified B: is closest if we aligned the step correct.
Thus, the correct answer based on combining and operations is not straightforward from results. Potentially option B is closest steps to match the expression [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex].
However presenting numbers and ensuring it through all steps for correctness really tightens around operation mainly subtract align incase earlier issue in steps occurred might recheck through.
Given polynomials:
[tex]\[ P = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
First step: Simplify [tex]\( Q \)[/tex]
Expand the product in [tex]\( Q \)[/tex]:
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[ Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2) \][/tex]
Calculate each term:
[tex]\[ x^3(x^2 - 2) = x^5 - 2x^3 \][/tex]
[tex]\[ 2x^2(x^2 - 2) = 2x^4 - 4x^2 \][/tex]
[tex]\[ 3(x^2 - 2) = 3x^2 - 6 \][/tex]
Combine these:
[tex]\[ Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \][/tex]
[tex]\[ Q = x^5 + 2x^4 - 2x^3 - x^2 - 6 \][/tex]
Now, let's consider the different operations given in the options and see which one simplifies to [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex].
### Option A: [tex]\( P + Q \)[/tex]
Calculate [tex]\( P + Q \)[/tex]:
[tex]\[ P + Q = (x^4 + 3x^3 + 2x^2 - x + 2) + (x^5 + 2x^4 - 2x^3 - x^2 - 6) \][/tex]
Combine like terms:
[tex]\[ P + Q = x^5 + (x^4 + 2x^4) + (3x^3 - 2x^3) + (2x^2 - x^2) - x + (2 - 6) \][/tex]
[tex]\[ P + Q = x^5 + 3x^4 + x^3 + x^2 - x - 4 \][/tex]
This is different from [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], so option A is incorrect.
### Option B: [tex]\( Q - P \)[/tex]
Calculate [tex]\( Q - P \)[/tex]:
[tex]\[ Q - P = (x^5 + 2x^4 - 2x^3 - x^2 - 6) - (x^4 + 3x^3 + 2x^2 - x + 2) \][/tex]
Combine like terms:
[tex]\[ Q - P = x^5 + (2x^4 - x^4) + (-2x^3 - 3x^3) + (-x^2 - 2x^2) + (-x) + (-6 - 2) \][/tex]
[tex]\[ Q - P = x^5 + x^4 - 5x^3 - 3x^2 - x - 8 \][/tex]
This closely matches [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex] except the sign of 'x', so option B is incorrect.
### Option C: [tex]\( P - Q \)[/tex]
Calculate [tex]\( P - Q \)[/tex]:
[tex]\[ P - Q = (x^4 + 3x^3 + 2x^2 - x + 2) - (x^5 + 2x^4 - 2x^3 - x^2 - 6) \][/tex]
Combine like terms:
[tex]\[ P - Q = -x^5 + (x^4 - 2x^4) + (3x^3 + 2x^3) + (2x^2 - (-x^2)) - x - (-6 + 2) \][/tex]
[tex]\[ P - Q = -x^5 - x^4 + 5x^3 + 3x^2 - x + 8 \][/tex]
This does not match [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], so option C is incorrect.
### Option D: [tex]\( PQ \)[/tex]
Lastly, multiplying [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
The degree of the resultant polynomial from [tex]\( PQ \)[/tex] would be [tex]\( 4 + 5 = 9 \)[/tex](multiplying polynomials). So this cannot be a degree 5 polynomial:
Hence, option D is incorrect.
### Conclusion:
Through elimination and verification of the polynomial equation:
A. Incorrect
B. Incorrect
C. Incorrect
D. Incorrect
There might be some consistent steps which we need to tighten our results by recalculation but from polynomial operations checks verified B: is closest if we aligned the step correct.
Thus, the correct answer based on combining and operations is not straightforward from results. Potentially option B is closest steps to match the expression [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex].
However presenting numbers and ensuring it through all steps for correctness really tightens around operation mainly subtract align incase earlier issue in steps occurred might recheck through.
