Question

Consider the polynomials given below.

P = x⁴ + 3x³ + 2x² - x + 2
Q = (x³ + 2x² + 3)(x² - 2)

Determine the operation that results in the simplified expression below.
x⁵ + x⁴ - 5x³ - 3x² + x - 8

A. Q - P
B. P - Q
C. PQ
D. P + Q

Answer 1

The operation that results in the simplified expression x⁵ + x⁴ - 5x³ - 3x² + x - 8 is A. Q - P.

Step-by-step explanation:

The operation that results in the simplified expression x⁵ + x⁴ - 5x³ - 3x² + x - 8 is A. Q - P.

First, let's simplify the given polynomials:

P = x⁴ + 3x³ + 2x² - x + 2

Q = (x³ + 2x² + 3)(x² - 2)

Multiplying Q by P gives:

PQ = (x⁴ + 3x³ + 2x² - x + 2) * (x³ + 2x² + 3)(x² - 2)

Expanding and combining like terms gives:

PQ = x⁷ + 2x⁶ + 5x⁵ + 3x⁴ - 2x³ + 11x² - 6x - 4

Subtracting PQ from the given expression gives:

x⁵ + x⁴ - 5x³ - 3x² + x - 8 - (x⁷ + 2x⁶ + 5x⁵ + 3x⁴ - 2x³ + 11x² - 6x - 4) = -x⁷ - 2x⁶ - 6x⁵ - 8x⁴ + 7x³ - 14x² + 7x - 4

Therefore, the operation that results in the simplified expression x⁵ + x⁴ - 5x³ - 3x² + x - 8 is A. Q - P.