What is the remainder when (3x⁴ + 2x³ - x² + 2x - 24) ÷ (x + 2)?

Solution:

Given, let f(x) = 3x⁴ + 2x³ - x² + 2x - 24

The expression (3x⁴ + 2x³ - x² + 2x - 24) is divided by (x+2).

The remainder theorem is stated as follows:

When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k).

Now, f(-2) = 3(-2)⁴ + 2(-2)³ - (-2)² + 2(-2) - 24

f(-2) = 3(16) + 2(-8) - 4 - 4 -24

f(-2) = 48 - 16 - 8 - 24

f(-2) = 48 - 24 - 24

f(-2) = 48 - 48

f(-2) = 0

Therefore, the remainder is 0.

What is the remainder when (3x⁴ + 2x³ - x² + 2x - 24) ÷ (x + 2)?

Summary:

The remainder when (3x⁴ + 2x³ - x² + 2x - 24) ÷ (x + 2) is 0.