So the remainder is **$ p(x) $** itself. But the problem asks for the remainder — which is a polynomial of degree less than 4 — so it is $ p(x) $, but we are to **report it in standard form**.

Sun, 03 May 2026 05:02:17 +0000

Understanding the Remainder: Reporting $ p(x) $ in Standard Polynomial Form When dividing polynomials, one key result from the division algorithm is that the remainder always has a degree strictly less than the divisor. In many problems, the remainder is described simply as $ p(x) $ itself — but to meet mathematical precision, especially in standardized testing or formal reporting, we must express this remainder in standard form: a polynomial expressed with terms arranged in descending order of degree, starting from the highest non-zero degree. Suppose $ p(x) $ was obtained as the remainder in a division operation where the divisor is a polynomial of degree 3 or higher. Then $ p(x) $ is guaranteed to have a degree less than 4 — meaning it can be written as: $$
p(x) = a_3x^3 + a_2x^2 + a_1x + a_0
$$