What is the remainder when 3x^4 + 2x^3 + x^2 + 2x + 24 is divided by 2?

To find the remainder when the polynomial 3x⁴ + 2x³ + x² + 2x + 24 is divided by 2, we can evaluate the polynomial at specific values that represent congruence classes modulo 2.

First, we look at the coefficients of the polynomial:

• The coefficient of x⁴ is 3, which is congruent to 1 modulo 2.
• The coefficient of x³ is 2, which is congruent to 0 modulo 2.
• The coefficient of x² is 1, which is congruent to 1 modulo 2.
• The coefficient of x is 2, which is congruent to 0 modulo 2.
• The constant term is 24, which is congruent to 0 modulo 2.

Now, we can rewrite the polynomial with these coefficients modulo 2:

P(x) ≡ 1x⁴ + 0x³ + 1x² + 0x + 0 (mod 2)

Thus, P(x) ≡ x⁴ + x² (mod 2).

Now, we want to evaluate this expression at x = 0 and x = 1 to find the remainders:

• If x = 0: P(0) ≡ 0⁴ + 0² ≡ 0 (mod 2)
• If x = 1: P(1) ≡ 1⁴ + 1² ≡ 1 + 1 ≡ 0 (mod 2)

Since both evaluations yield a result of 0, the remainder when dividing the polynomial by 2 is:

Remainder: 0