The student's question is about proving that within the factor ring Z[x]/I, the polynomial (x⁵+2x³-x²+x+4)+I is equivalent to 5+I. By replacing x² with -1, the polynomial simplifies to 5, demonstrating the required equivalence.
The student is asking to show that in the factor ring Z[x]/I where I=➩x²+1⟪, the equivalence of (x⁵+2x³−x²+x+4)+I to 5+I holds. In the ring Z[x], elements of I are all multiples of x²+1, so when taking any polynomial f(x) in Z[x] modulo I, we can replace occurrences of x² with −1 because x²−1 is in I and thus equivalent to 0 in the factor ring.
To simplify (x⁵+2x³−x²+x+4) in Z[x]/I, we replace x² with −1, which gives us x⁵+2x³−(−1)+x+4 = x⁵+2x(−1)+(−1)+x+4 = x⁵−2x+5. Because x is arbitrary, x⁵ is also essentially −1, so the polynomial reduces to −2(−1)+5 = 5, which means (x⁵+2x³−x²+x+4)+I is indeed equivalent to 5+I.