2.3.4 Polynomial Division and Remainder Theorem

Performing polynomial long division and synthetic division, and applying the Remainder and Factor Theorems.

定义

Polynomial Division and Remainder Theorem refers to the process of dividing one polynomial by another and determining the quotient and remainder. When a polynomial \(P(x)\) is divided by a divisor polynomial \(D(x)\), we can express this relationship as \(P(x) = D(x) \cdot Q(x) + R(x)\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder with degree less than the degree of \(D(x)\). The Remainder Theorem states that when a polynomial \(P(x)\) is divided by a linear divisor \((x - a)\), the remainder is equal to \(P(a)\). The Factor Theorem is a special case of the Remainder Theorem: \((x - a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). Polynomial division can be performed using two main methods: polynomial long division (similar to numerical long division) and synthetic division (a more efficient method for dividing by linear factors of the form \((x - a)\)).

核心公式

• \(P(x) = D(x) \cdot Q(x) + R(x)\)
• \(P(a) = R \text{ (Remainder Theorem)}\)
• \((x - a) \text{ is a factor of } P(x) \iff P(a) = 0 \text{ (Factor Theorem)}\)
• \(\text{If } P(x) = (x - a) \cdot Q(x) + R, \text{ then } \deg(R) < \deg(x - a) = 1\)
• \(P(x) = (x - a)^n \cdot Q(x) + R(x) \text{ where } \deg(R) < n\)

易错点

• ⚠️ Forgetting that the remainder must have a degree strictly less than the divisor's degree; students sometimes incorrectly continue dividing when the remainder should stop
• ⚠️ Misapplying the Remainder Theorem by evaluating \(P(x)\) at the wrong value; for example, when dividing by \((x + 3)\), students must evaluate \(P(-3)\) not \(P(3)\)
• ⚠️ Making arithmetic errors in synthetic division, particularly with sign changes or when the leading coefficient of the divisor is not 1
• ⚠️ Confusing the Factor Theorem with the Remainder Theorem; not recognizing that a factor exists only when the remainder equals zero, not just any non-zero remainder