📚 Polynomial Theorems | 多项式定理
Understanding polynomial theorems is fundamental to mastering algebra in IB Mathematics. These theorems provide efficient tools for factorising polynomials, finding roots, and solving higher-degree equations. This article covers the key theorems you need, with clear explanations and examples.
理解多项式定理是掌握 IB 数学代数的基石。这些定理为因式分解多项式、求根以及解高次方程提供了高效的工具。本文将涵盖你需要掌握的关键定理，并配以清晰的解释和示例。
1. Polynomial Basics | 多项式基础
A polynomial in one variable x is an expression of the form anxⁿ + an-1xⁿ⁻¹ + ... + a1x + a0, where n is a non-negative integer and each coefficient ai is a real number (or complex). The degree of the polynomial is the highest power n with an ≠ 0.
一元多项式 x 的表达式形式为 anxⁿ + an-1xⁿ⁻¹ + ... + a1x + a0，其中 n 是非负整数，每个系数 ai 是实数（或复数）。多项式的次数是最高次幂 n，且首项系数 an ≠ 0。
The term an is the leading coefficient, and a0 is the constant term. If an = 1, the polynomial is called monic.
an 称为首项系数，a0 为常数项。若 an = 1，该多项式称为首一多项式。
2. Polynomial Long Division | 多项式长除法
tient q(x) and a remainder r(x) of degree less than d(x), so that f(x) = d(x)·q(x) + r(x).
在学习定理之前，你必须熟练多项式除法。如同数值长除法一样，将多项式 f(x) 除以除式 d(x) 时，反复消去首项。过程得到商式 q(x) 和余式 r(x)，其次数低于 d(x)，满足 f(x) = d(x)·q(x) + r(x)。
nt of 2x² + 5x + 0 and a remainder of 4. Hence 2x³ + 3x² − 5x + 4 = (x − 1)(2x² + 5x) + 4.
例如，2x³ + 3x² − 5x + 4 除以 x − 1，商为 2x² + 5x + 0，余数为 4。因此 2x³ + 3x² − 5x + 4 = (x − 1)(2x² + 5x) + 4。
3. The Remainder Theorem | 余数定理
The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x − a), the remainder is simply f(a). This is a powerful shortcut: you do not need to perform the full division to find the remainder.
余数定理指出，当多项式 f(x) 除以一次因式 (x − a) 时，余数就是 f(a)。这是一个强有力的捷径：你无需完整执行除法就能求出余数。
3x² + 2x − 5 divided by (x − 2), calculate f(2) = 16 − 12 + 4 − 5 = 3. The remainder is 3.
例如，求 f(x) = x⁴ − 3x² + 2x − 5 除以 (x − 2) 的余数，只需计算 f(2) = 16 − 12 + 4 − 5 = 3，余数为 3。
If the divisor is of the form (ax − b), rewrite it as a(x − b/a) and apply the theorem: the remainder is f(b/a).
若除式为 (ax − b)，可写为 a(x − b/a) 形式，应用定理：余数为 f(b/a)。
4. The Factor Theorem | 因式定理
factor of f(x) if and only if f(a) = 0. That is, a is a root (or zero) of the polynomial.
因式定理是余数定理的直接推论。它指出 (x − a) 是 f(x) 的因式当且仅当 f(a) = 0。也就是说，a 是多项式的一个根（或零点）。
ers: f(1) = 0, so (x − 1) is a factor. Use division to find the other factor and continue.
该定理将求根转化为寻找因式。例如，要对 f(x) = x³ − 2x² − 5x + 6 因式分解，检验小整数：f(1) = 0，因此 (x − 1) 是一个因式。用除法找到其他因式，继续分解。
5. Synthetic Division (Horner's Method) | 综合除法（霍纳法）
Synthetic division is an efficient algorithm for dividing a polynomial by a linear factor (x − a). It uses only the coefficients and reduces writing. The procedure involves bringing down the leading coefficient, then repeatedly multiplying by a and adding the next coefficient. The last number obtained is the remainder.
综合除法是将多项式除以一次因式 (x − a) 的高效算法，只用系数计算，减少书写。步骤为：落下首项系数，然后反复乘以 a 并加上下一个系数。最后得到的数即为余数。
ly 2×2 = 4, add to 5 gives 9. Remainder = 9, quotient coefficients: 2, 1, 2 → 2x² + x + 2.
例如，2x³ − 3x² + 0x + 5 除以 (x − 2)。系数列为：2, -3, 0, 5。落下 2，2×2 = 4 加上 -3 得 1；1×2 = 2 加上 0 得 2；2×2 = 4 加上 5 得 9。余数为 9，商的系数为 2, 1, 2，即 2x² + x + 2。
| 2 | -3 | 0 | 5 |
| 4 | 2 | 4 | |
| 2 | 1 | 2 | 9 |
The first row is coefficients, second row shows the products added, and the third row gives the quotient coefficients and remainder (9). Synthetic division is especially useful when combined with the Factor Theorem to repeatedly factor out roots.
第一行是系数，第二行是相加的乘积，第三行给出商的系数及余数（9）。综合除法与因式定理结合使用时特别有用，可反复分解出根。
6. The Rational Root Theorem | 有理根定理
ossible rational roots of a polynomial with integer coefficients. If a polynomial f(x) = anxⁿ + ... + a0 has a rational root p/q in lowest terms, then p must be a factor of the constant term a0, and q must be a factor of the leading coefficient an.
有理根定理帮助你找出整数系数多项式可能的有理根。若多项式 f(x) = anxⁿ + ... + a0 有一个最简分数形式的有理根 p/q，则 p 必为常数项 a0 的因数，q 必为首项系数 an 的因数。
ded by factors of 2: ±1, ±2, ±3, ±6, ±½, ±3/2. Testing these will locate the actual roots.
对于 f(x) = 2x³ − 3x² − 11x + 6，可能的有理根为 ±6 的因数除以 2 的因数：±1, ±2, ±3, ±6, ±½, ±3/2。逐一检验可找到真正的根。
Note that the theorem lists only possibilities—some may not be roots. Use substitution or synthetic division to test them. It works for any polynomial with integer coefficients.
注意该定理仅列出可能值——有些可能不是根。用代入法或综合除法检验它们。该定理适用于任何整系数多项式。
7. Vieta's Formulas: Sum and Product of Roots | 韦达定理：根的和与积
Vieta' s. For a quadratic ax² + bx + c = 0 with roots α and β, we have α + β = −b/a and αβ = c/a.
韦达定理将多项式的系数与根的对称和联系起来。对于二次方程 ax² + bx + c = 0，根为 α 和 β，有 α + β = −b/a，αβ = c/a。
For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ:
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ：
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
These relations are extremely useful when you know some roots and need to find others, or when constructing a polynomial from given roots.
当你已知部分根需要求其他根，或根据给定根构造多项式时，这些关系极为有用。
8. Complex Roots and Conjugate Pairs | 复数根与共轭对
The Fundamental Theorem of Algebra guarantees that a polynomial of degree n with real (or complex) coefficients has exactly n roots in the complex number system, counting multiplicities. If all coefficients are real, any non-real complex roots must occur in conjugate pairs. That is, if a + bi is a root, then a − bi is also a root.
代数基本定理确保了次数为 n 的实系数（或复系数）多项式在复数系中恰有 n 个根（计重数）。若所有系数均为实数，任何非实复数根必定以共轭对形式出现。即若 a + bi 是一个根，则 a − bi 也是根。
−i. The cubic x³ − x² + x − 1 has a real root x = 1 and a complex conjugate pair i and −i.
例如，多项式 x² + 1 没有实数根，但有两个复数根 i 和 −i。三次式 x³ − x² + x − 1 有一个实数根 x = 1，以及共轭复数对 i 和 −i。
Thus, if you discover one complex root of a real polynomial, you immediately know another, which helps in factorisation.
因此，一旦你找到实系数多项式的一个复数根，你便立即知道另一个根，这有助于因式分解。
9. Solving Higher-Degree Equations | 求解高次方程
Combining the Factor Theorem, synthetic division, and the Rational Root Theorem gives a systematic method for solving polynomial equations of degree 3 or higher. Start by listing possible rational roots, test them until you find one, then use synthetic division to reduce the degree. Repeat the process on the quotient polynomial.
结合因式定理、综合除法和有理根定理，可得到求解三次或更高次多项式方程的系统方法。首先列出可能的有理根，逐一检验直至找到一根，然后用综合除法降次。对商多项式重复这一过程。
the quadratic: (2x−1)(x+2) = 0, so x = ½, x = −2. The complete solution set is {3, ½, −2}.
例：求解 2x³ − 3x² − 11x + 6 = 0。可能有理根：±1, ±2, ±3, ±6, ±½, ±3/2。检验 x = 1：2−3−11+6 = −6，不为零。检验 x = −1：−2−3+11+6 = 12，不成立。检验 x = 2：16−12−22+6 = −12。检验 x = 3：54−27−33+6 = 0，故 x = 3 为一个根。用 (x−3) 综合除法得 2x² + 3x − 2。分解二次式：(2x−1)(x+2) = 0，故 x = ½, x = −2。完全解集为 {3, ½, −2}。
10. Irrational Root Conjugates (Extension) | 无理根共轭（拓展）
If a polynomial has rational coefficients and an irrational root of the form p + √q (where √q is irrational), its conjugate p − √q is also a root. This works similarly to complex conjugates and arises from the quadratic formula.
若多项式系数为有理数，且有一个形如 p + √q 的无理根（√q 为无理数），则其共轭 p − √q 也是根。这与复数共轭类似，源自二次公式。
For example, the polynomial x² − 4x + 1 has roots 2 + √3 and 2 − √3. The theorem can help identify missing roots when factoring polynomials with irrational solutions.
例如，多项式 x² − 4x + 1 的根为 2 + √3 和 2 − √3。该定理可在分解有无理解的因式时帮助识别缺失的根。
11. Common Mistakes and Exam Tips | 常见错误与应试技巧
Mistake 1: ve factors when applying the Rational Root Theorem. Always list possibilities with ± sign.
错误一：应用有理根定理时忘记考虑正负因数。务必列出带 ± 的所有可能值。
Mistake 2: Misapplying the Remainder Theorem when the divisor is not monic, e.g., (2x−1). You must compute f(½), not f(−1) or f(1).
错误二：当除式不是首一式时，如 (2x−1)，余数定理用错。必须计算 f(½)，而不是 f(−1) 或 f(1)。
Mistake 3: Confusing the signs in Vieta' = −b/a, product = c/a (for quadratic). The alternating signs continue for higher degrees.
错误三：混淆韦达定理中的符号。记住：根的和 = −b/a，积 = c/a（二次式）。符号交错延续至更高次。
Exam Tip: Always verify your factorisation by expanding. Use the constant term to quickly check if the product of roots matches.
应试技巧：始终通过展开验证因式分解结果。利用常数项快速检查根之积是否吻合。
12. Summary | 总结
Polynomial theorems are interconnected tools that simplify working with polynomials. The Remainder and Factor Theorems give meaning to function evaluation, synthetic division speeds up calculations, the Rational Root Theorem guides root searching, and Vieta's formulas unlock symmetric relations. Master these, and polynomial equations become manageable.
多项式定理是一套相互关联的工具，可简化多项式的处理。余数定理和因式定理赋予函数求值以意义，综合除法提高计算速度，有理根定理指引求根方向，韦达定理揭示对称关系。掌握这些，多项式方程便迎刃而解。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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