If $-2x^3 + 6x^2 + 17x$ - 21 is divided by $(x + 1)$, then the remainder is

Analyze the question: We need the remainder when a polynomial is divided by (x + 1). This is a job for the Remainder Theorem.

Concept: When a polynomial f(x) is divided by (x − a), the remainder is f(a). For (x + 1), we have a = −1, so the remainder is f(−1).

Solution:

f(x) = −2x³ + 6x² + 17x − 21

f(−1) = −2(−1)³ + 6(−1)² + 17(−1) − 21

• −2(−1)³ = −2(−1) = +2
• 6(−1)² = 6(1) = 6
• 17(−1) = −17
• Constant = −21

f(−1) = 2 + 6 − 17 − 21 = 8 − 38 = −30

Check with synthetic division (using −1):

Coefficients: −2, 6, 17, −21

• Bring down −2
• −2 × −1 = 2; 6 + 2 = 8
• 8 × −1 = −8; 17 − 8 = 9
• 9 × −1 = −9; −21 − 9 = −30

The remainder is −30, which matches.

Answer: The remainder is −30, which is option C.

Tip: For a divisor (x + a), substitute x = −a. Be careful with signs when substituting a negative number into odd powers, since (−1)³ = −1 but (−1)² = +1. Most mistakes here come from sign errors.