Question

select the correct answer.
consider the polynomials given below.
$p = x^4 + 3x^3 + 2x^2 - x + 2$
$q = (x^3 + 2x^2 + 3)(x^2 - 2)$
determine the operation that results in the simplified expression below:
$x^5 + x^4 - 5x^3 - 3x^2 + x - 8$
a. $pq$
b. $p - q$
c. $p + q$
d. $q - p$

Explanation:

Step1: Expand Q

First, expand \( Q=(x^{3}+2x^{2}+3)(x^{2}-2) \). Using the distributive property (FOIL for polynomials):

$$
LATEXBLOCK0
$$

Step2: Compute Q - P

Now, compute \( Q - P \) where \( P = x^{4}+3x^{3}+2x^{2}-x + 2 \):

$$
LATEXBLOCK1
$$

This matches the given simplified expression. Let's check other options briefly:

• For \( PQ \), the degree would be \( 4 + 5 = 9 \) (since \( P \) is degree 4 and \( Q \) is degree 5), but the given expression is degree 5, so A is wrong.
• For \( P - Q \), the leading term would be \( -x^{5} \) (since \( P \) has no \( x^{5} \) term), which doesn't match.
• For \( P + Q \), the leading term is \( x^{5} \), but combining like terms would give \( x^{5}+3x^{4}+x^{3}+x^{2}-x - 4 \), which doesn't match.

Answer:

D. \( Q - P \)