Use the Remainder Theorem quickly Find the remainder when f(x) = x^4 − 6x^3 + 2x^2 + 3x + 1 is divided by (x − 2).

Introduction / Context:The Remainder Theorem states that the remainder when a polynomial f(x) is divided by (x − a) equals f(a). This gives a fast, error-resistant method.

Given Data / Assumptions:

• f(x) = x^4 − 6x^3 + 2x^2 + 3x + 1.
• Divisor is (x − 2), so a = 2.

Concept / Approach:Compute f(2) directly and that value is the remainder.

Step-by-Step Solution:f(2) = 2^4 − 6·2^3 + 2·2^2 + 3·2 + 1.= 16 − 6·8 + 2·4 + 6 + 1.= 16 − 48 + 8 + 6 + 1 = −17.

Verification / Alternative check:Synthetic division at 2 also yields remainder −17.

Why Other Options Are Wrong:12, 17, −12, −5 arise from partial sums or sign slips; only −17 equals f(2).

Common Pitfalls:Forgetting order of operations; miscomputing 6·2^3 as 12 or 48 incorrectly.

Final Answer:−17