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Consider the polynomials given below.

[tex]\[
\begin{array}{l}
P(x)=x^4+3x^3+2x^2-x+2 \\
Q(x)=\left(x^3+2x^2+3\right)\left(x^2-2\right)
\end{array}
\][/tex]

Determine the operation that results in the simplified expression below.

[tex]\[
x^5 + x^4 - 5x^3 - 3x^2 + x - 8
\][/tex]

A. [tex]\(P-Q\)[/tex]
B. [tex]\(Q-P\)[/tex]
C. [tex]\(P+Q\)[/tex]
D. [tex]\(PQ\)[/tex]


Sagot :

Let's break down the problem and determine which operation on the polynomials [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] results in the given simplified expression.

First, let's rewrite the polynomials:
[tex]\[ P(x) = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q(x) = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]

Next, we'll expand [tex]\( Q(x) \)[/tex]:
[tex]\[ Q(x) = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
[tex]\[ Q(x) = x^3 \cdot x^2 + x^3 \cdot (-2) + 2x^2 \cdot x^2 + 2x^2 \cdot (-2) + 3 \cdot x^2 + 3 \cdot (-2) \][/tex]
[tex]\[ Q(x) = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \][/tex]
[tex]\[ Q(x) = x^5 + 2x^4 - 2x^3 - x^2 - 6 \][/tex]

Now let's examine the given simplified expression:
[tex]\[ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]

We need to compare this expression against the results of performing various operations on [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].

Let's consider each operation one by one.

Operation: [tex]\( P(x) - Q(x) \)[/tex]

[tex]\[ P(x) - Q(x) = \left( x^4 + 3x^3 + 2x^2 - x + 2 \right) - \left( x^5 + 2x^4 - 2x^3 - x^2 - 6 \right) \][/tex]
[tex]\[ P(x) - Q(x) = -x^5 + x^4 + 5x^3 + 3x^2 - x + 8 \][/tex]

Operation: [tex]\( Q(x) - P(x) \)[/tex]

[tex]\[ Q(x) - P(x) = \left( x^5 + 2x^4 - 2x^3 - x^2 - 6 \right) - \left( x^4 + 3x^3 + 2x^2 - x + 2 \right) \][/tex]
[tex]\[ Q(x) - P(x) = x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]

Given the comparison, it is clear that the correct operation is:
[tex]\[ Q(x) - P(x) \][/tex]

Thus, the operation that results in the simplified expression:

[tex]\[ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]

is
[tex]\[ \boxed{Q - P} \][/tex]

The answer is:
B. [tex]\( Q - P \)[/tex]