Polynomials | 多项式

📚 Polynomials | 多项式

A polynomial is one of the most fundamental algebraic structures in IB Mathematics. Whether you are analysing graphs, solving equations, or applying the Factor Theorem, polynomials provide a powerful toolkit for modelling and problem-solving. This revision guide walks you through the key concepts, techniques, and theorems that are assessed in IB Mathematics, from basic operations to Vieta’s formulas and beyond.

多项式是 IB 数学中最基本的代数结构之一。无论是分析图像、解方程还是应用因式定理，多项式都提供了强大的建模和解题工具。本复习指南将带你梳理 IB 数学中考查的核心概念、解题技巧和重要定理，从基本运算到韦达定理，一应俱全。

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1. What is a Polynomial? | 什么是多项式？

A polynomial in one variable x is an expression of the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non-negative integer, the coefficients a₀, a₁, …, aₙ are real numbers (often rational or integer in IB problems), and aₙ ≠ 0. The powers of x must be whole numbers, so terms like 3x⁻² or 5√x are not allowed in a polynomial.

一个关于变量 x 的多项式是形如 P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 是非负整数，系数 a₀, a₁, …, aₙ 为实数（IB 题目中常为有理数或整数），且 aₙ ≠ 0。x 的指数必须是整数，因此像 3x⁻² 或 5√x 这样的项不能出现在多项式中。

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2. Degree, Leading Coefficient and Standard Form | 次数、首项系数与标准形式

The degree of a polynomial is the highest exponent of the variable. The leading coefficient is the coefficient of the term with the highest degree. Writing a polynomial in descending powers of x is called standard form. For example, P(x) = 7x⁴ – 5x³ + 2x – 9 has degree 4 and leading coefficient 7. A constant polynomial has degree 0, and the zero polynomial has undefined degree.

多项式的次数是变量指数的最大值。首项系数是次数最高项对应的系数。按 x 的降幂排列书写多项式称为标准形式。例如，P(x) = 7x⁴ – 5x³ + 2x – 9 的次数是 4，首项系数是 7。常数多项式的次数为 0，零多项式的次数未定义。

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3. Adding, Subtracting and Multiplying Polynomials | 多项式的加减与乘法

To add or subtract polynomials, simply combine like terms – terms that have exactly the same variable part. Multiplication uses the distributive property: multiply each term of the first polynomial by each term of the second and then collect like terms. A common IB trap is forgetting to distribute a minus sign when subtracting; always place brackets around the subtracted polynomial.

多项式的加减只需合并同类项，即含有完全相同变量部分的项。乘法运用分配律：将第一个多项式的每一项与第二个多项式的每一项相乘，再合并同类项。IB 中常见的一个易错点是做减法时忘记对减号后面的多项式加上括号，一定要先括起来再处理符号。

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4. Polynomial Long Division | 多项式长除法

Polynomial long division mirrors the long division of numbers. To divide P(x) by D(x), write them in descending powers, divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by that result, subtract, bring down the next term, and repeat. The final expression is P(x) = D(x) × Q(x) + R(x), where the degree of R(x) is less than the degree of D(x).

多项式长除法与数的长除法非常相似。将被除式 P(x) 与除式 D(x) 按降幂排列，用被除式的首项除以除式的首项，将结果与整个除式相乘，相减，移下下一项，并重复此过程。最终表达式为 P(x) = D(x) × Q(x) + R(x)，其中余式 R(x) 的次数低于除式的次数。

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5. Synthetic Division | 综合除法

Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form x – c. Write the coefficients of P(x) in a row, bring down the leading coefficient, multiply by c and add to the next coefficient, repeating until the end. The last number is the remainder R, and the earlier numbers are coefficients of the quotient Q(x), which has degree one less than P(x). Synthetic division is faster but only works for linear divisors.

综合除法是多项式除以形如 x – c 的一次因式的简便算法。将 P(x) 的系数排成一行，把首项系数直接下移，然后乘以 c 加到下一个系数上，重复直至末尾。最后一个数是余数 R，前面的数便是商式 Q(x) 的系数，其次数比 P(x) 低一次。综合除法速度更快，但只适用于一次除式。

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6. The Remainder Theorem | 余数定理

The Remainder Theorem states that when a polynomial P(x) is divided by x – c, the remainder is simply P(c). This powerful result allows you to evaluate P(c) either by substitution or by synthetic division. For instance, if P(x) = 2x³ – 5x² + 3x – 8 and we divide by x – 2, the remainder is P(2) = 2(8) – 5(4) + 6 – 8 = –6. No long division is needed.

余数定理指出，当多项式 P(x) 除以 x – c 时，所得余数就是 P(c)。利用这个强大的结论，你可以通过代入或综合除法直接求值。例如，若 P(x) = 2x³ – 5x² + 3x – 8 除以 x – 2，余数即为 P(2) = 2(8) – 5(4) + 6 – 8 = –6，完全不需要长除法。

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7. The Factor Theorem and Its Use | 因式定理及其应用

The Factor Theorem is a direct consequence of the Remainder Theorem: x – c is a factor of P(x) if and only if P(c) = 0. This gives a systematic method for finding factors: test potential values of c (try ±1, ±2, etc.) until a zero is found, then divide to reduce the polynomial. In IB questions, this is often the first step in factorising cubic or quartic polynomials and solving equations.

因式定理是余数定理的直接推论：x – c 是 P(x) 的因式当且仅当 P(c) = 0。这为寻找因式提供了系统的方法：尝试可能的 c 值（如 ±1, ±2 等），找到零点后做除法将多项式降次。在 IB 的考题中，这往往是对三次或四次多项式进行因式分解并解方程的第一步。

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8. Factoring Polynomials | 多项式的因式分解

Factoring polynomials is essential for solving equations and simplifying expressions. Common techniques include: taking out the greatest common factor; grouping terms; recognising special products such as difference of squares a² – b² = (a – b)(a + b), sum and difference of cubes a³ ± b³ = (a ± b)(a² ∓ ab + b²); and applying the Factor Theorem. For quadratics, inspection, completing the square, or the quadratic formula can also help reveal factors with irrational or complex roots.

多项式的因式分解是解方程和化简表达式的关键。常用的方法包括：提取公因式、分组分解、识别乘法公式如平方差 a² – b² = (a – b)(a + b)、立方和与立方差 a³ ± b³ = (a ± b)(a² ∓ ab + b²)，以及应用因式定理。对于二次式，还可以通过十字相乘法、配方法或求根公式得到含有无理根或复数根的因式。

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9. Solving Polynomial Equations | 解多项式方程

To solve a polynomial equation P(x) = 0, first move all terms to one side so that the equation equals zero. Factorise as completely as possible, then set each factor equal to zero. The Zero Product Property guarantees that if a × b = 0, then a = 0 or b = 0. For higher-degree equations, repeated use of the Factor Theorem and division reduces the degree until you obtain linear and quadratic factors. Always check for repeated roots and state their multiplicity where required.

解多项式方程 P(x) = 0 时，先将所有项移到一边使方程等于零。尽可能彻底地进行因式分解，然后令每一个因式等于零。零乘积性质保证如果 a × b = 0，则 a = 0 或 b = 0。对于高次方程，重复使用因式定理和除法降次，直至得到一次和二次因式。永远要检查重根并在需要时标出其重数。

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10. Graphs of Polynomials: End Behaviour and Turning Points | 多项式图像：首尾趋势与极值点

The graph of a polynomial function is a smooth, continuous curve. The leading coefficient and degree determine the end behaviour: if the degree is even and leading coefficient positive, both ends rise; even degree with negative coefficient → both ends fall; odd degree with positive coefficient → falls left, rises right; odd degree with negative coefficient → rises left, falls right. The maximum number of turning points is one less than the degree. Understanding these features helps you sketch graphs and interpret real-life models.

多项式函数的图像是一条光滑且连续的曲线。首项系数和次数决定了首尾趋势：若次数为偶数且首项系数为正，两端均向上延伸；偶次且系数为负则两端向下；奇次且系数为正时左端向下、右端向上；奇次且系数为负则左端向上、右端向下。极值点（拐点）的最大个数为次数减一。理解这些特征有助于你快速绘制草图并理解实际问题的建模。

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11. Vieta’s Formulas: Linking Roots and Coefficients | 韦达定理：根与系数的联系

For a quadratic ax² + bx + c = 0 with roots r₁ and r₂, Vieta’s formulas give r₁ + r₂ = –b/a and r₁ × r₂ = c/a. For a cubic ax³ + bx² + cx + d = 0 with roots r₁, r₂, r₃, the relationships are: sum of roots = –b/a, sum of pairwise products = c/a, and product of roots = –d/a. These formulas appear frequently in IB proofs and problems where you are asked to find a polynomial given its roots or to determine unknown coefficients.

对于二次方程 ax² + bx + c = 0，设两根为 r₁, r₂，韦达定理给出 r₁ + r₂ = –b/a 和 r₁ × r₂ = c/a。对于三次方程 ax³ + bx² + cx + d = 0，设三根为 r₁, r₂, r₃，则有根的和 = –b/a，两两乘积之和 = c/a，根的乘积 = –d/a。这些关系经常出现在 IB 的证明题中，或者已知根反求多项式以及确定未知系数的题目里。

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12. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Be meticulous with signs, especially when subtracting polynomials or applying synthetic division with a negative root. Always check whether a value is a zero before assuming a factor; a quick P(c) calculation can save time. When dividing, write missing terms explicitly with zero coefficients to avoid misalignment. For graph sketching, label intercepts and indicate end behaviour rather than trying to draw a perfect scale plot. Finally, if you spot a quotient that is quadratic and cannot factor over real numbers, use the discriminant to describe the nature of the remaining roots.

要格外留意符号，特别是在多项式减法或使用含有负根的综合除法时。在认定一个值是否为因式之前，先快速计算 P(c) 可以省下不少时间。做除法时，要将缺项的系数用零补全，以免错位。画多项式草图时，标出截距并反映首尾趋势即可，没必要追求精确的比例尺。最后，如果所得商式为二次式且在实数范围内无法分解，可用判别式说明其余根的性质。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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