2.3.3 Roots and Zeros of Polynomials¶

Finding polynomial roots using the Rational Root Theorem, synthetic division, and understanding multiplicity of zeros.

定义¶

A root (or zero) of a polynomial \(P(x)\) is a value \(r\) such that \(P(r) = 0\). The Rational Root Theorem states that if a polynomial \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) with integer coefficients has a rational root \(\frac{p}{q}\) (in lowest terms), then \(p\) divides the constant term \(a_0\) and \(q\) divides the leading coefficient \(a_n\). The multiplicity of a zero is the number of times the corresponding factor appears in the complete factorization of the polynomial. A zero with multiplicity \(m\) means the factor \((x - r)\) appears \(m\) times in the factorization. Synthetic division is an efficient method for dividing a polynomial by a linear factor \((x - c)\) and for evaluating \(P(c)\) using the Remainder Theorem, which states that the remainder when \(P(x)\) is divided by \((x - c)\) equals \(P(c)\).

核心公式¶

\(P(x) = (x - r_1)(x - r_2) \cdots (x - r_n) + \text{remainder}\)
\(P(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2} \cdots (x - r_k)^{m_k}\)
\(\text{If } P(x) = Q(x)(x - c) + R, \text{ then } R = P(c)\)
\(\text{Rational Root Theorem: possible rational roots} = \pm \frac{\text{factors of } a_0}{\text{factors of } a_n}\)
\(P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \text{ has at most } n \text{ real roots}\)

易错点¶

⚠️ Forgetting to consider both positive and negative factors when applying the Rational Root Theorem—students often only test positive rational roots and miss negative ones
⚠️ Confusing multiplicity with the number of distinct roots—a polynomial of degree \(n\) has exactly \(n\) roots (counting multiplicity), not necessarily \(n\) distinct roots
⚠️ Incorrectly performing synthetic division by not properly aligning coefficients or forgetting to include zero coefficients for missing terms in the polynomial
⚠️ Misinterpreting the behavior of a polynomial at a zero with even multiplicity (touches but doesn't cross the x-axis) versus odd multiplicity (crosses the x-axis)