Factor Theorem and Its Applications | 因式定理及其应用

📚 Factor Theorem and Its Applications | 因式定理及其应用

The Factor Theorem is a fundamental result in algebra that links the roots of a polynomial with its linear factors. It states that for a polynomial f(x), if f(a) = 0, then (x – a) is a factor of f(x). This theorem is widely used to factorise polynomials, solve equations, and analyse polynomial functions.

因式定理是代数学中的基本结论，它将多项式的根与一次因式联系起来。它指出：对于多项式 f(x)，若 f(a)=0，则 (x-a) 是 f(x) 的一个因式。该定理广泛用于因式分解多项式、解方程和分析多项式函数。

1. The Factor Theorem Statement | 因式定理的表述

If f(x) is a polynomial of degree n ≥ 1, and a is any real or complex number, then (x – a) is a factor of f(x) if and only if f(a) = 0. This means that evaluating the polynomial at x = a gives zero exactly when the linear factor x – a divides the polynomial without remainder.

如果 f(x) 是次数 n ≥ 1 的多项式，a 是任意实数或复数，那么 (x-a) 是 f(x) 的因式当且仅当 f(a)=0。也就是说，当且仅当一次因式 x-a 整除多项式且余数为零时，多项式在 x=a 处的值为零。

Example: For f(x) = x² – 5x + 6, f(2) = 4 – 10 + 6 = 0, so (x – 2) is a factor. Indeed, x² – 5x + 6 = (x – 2)(x – 3).

示例：对于 f(x) = x² – 5x + 6，有 f(2) = 4 – 10 + 6 = 0，因此 (x-2) 是一个因式。事实上，x² – 5x + 6 = (x – 2)(x – 3)。

2. Proof of the Factor Theorem | 因式定理的证明

By the division algorithm, any polynomial f(x) can be divided by (x – a) to give a quotient Q(x) and a constant remainder R. Since x – a has degree 1, the remainder is a constant, and by the Remainder Theorem this remainder equals f(a).

根据带余除法，任何多项式 f(x) 除以 (x-a) 都会得到商式 Q(x) 与常数余项 R。由于 x-a 是一次式，余项为常数，而根据余数定理，该余项等于 f(a)。

If f(a) = 0, then the remainder R = 0, so the division is exact and (x – a) is a factor. Conversely, if (x – a) is a factor, then f(x) = (x – a)Q(x), and substituting x = a gives f(a) = 0.

若 f(a) = 0，则余项 R = 0，因此除法为整除，(x-a) 是因式。反过来，若 (x-a) 是因式，则 f(x) = (x-a)Q(x)，把 x=a 代入得 f(a)=0。

3. Relationship with the Remainder Theorem | 与余数定理的关系

The Remainder Theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a). The Factor Theorem is simply a special case of this result: the remainder is zero, so the divisor is a factor.

余数定理指出：多项式 f(x) 除以 (x-a) 所得余数为 f(a)。因式定理正是该结论的特例：当余数为零时，除式就是因式。

This relationship allows us to test possible linear factors quickly. Instead of performing long division, we only need to evaluate f(a) at a candidate value a.

这种关系使我们能够快速检验可能的一次因式。我们只需在候选值 a 处计算 f(a)，而不必进行长除法。

4. Using the Factor Theorem to Test Factors | 用因式定理检验因式

To test whether (x – a) is a factor, substitute a into the polynomial and check whether the result is zero. For a polynomial with integer coefficients, possible rational roots are of the form ±p/q, where p divides the constant term and q divides the leading coefficient.

要检验 (x-a) 是否为因式，只需将 a 代入多项式，检查结果是否为零。对于整数系数的多项式，可能的有理根形如 ±p/q，其中 p 整除常数项，q 整除首项系数。

Consider f(x) = 2x³ – 3x² – 11x + 6. The possible rational roots are ±1, ±2, ±3, ±6, ±½, ±3/2. Testing x = 2 gives f(2) = 16 – 12 – 22 + 6 = -12, so (x – 2) is not a factor. Testing x = -2 gives f(-2) = -16 – 12 + 22 + 6 = 0, so (x + 2) is a factor.

考虑 f(x) = 2x³ – 3x² – 11x + 6。可能有理根为 ±1、±2、±3、±6、±½、±3/2。检验 x=2 得 f(2) = 16 – 12 – 22 + 6 = -12，因此 (x-2) 不是因式。检验 x=-2 得 f(-2) = -16 – 12 + 22 + 6 = 0，所以 (x+2) 是因式。

5. Combining with Synthetic Division | 综合除法的结合使用

After finding one linear factor, synthetic division can reduce the polynomial to a lower-degree quotient. This makes further factorisation much easier.

在找到一个一次因式后，综合除法可将原多项式降为较低次数的商式，这使后续分解更加容易。

For f(x) = 2x³ – 3x² – 11x + 6, since (x + 2) is a factor, divide the polynomial by (x + 2) to get 2x² – 7x + 3. This quotient can then be factorised as (2x – 1)(x – 3). Hence f(x) = (x + 2)(2x – 1)(x – 3).

对于 f(x) = 2x³ – 3x² – 11x + 6，因为 (x+2) 是因式，除以 (x+2) 得 2x² – 7x + 3。该商式可分解为 (2x – 1)(x – 3)。因此 f(x) = (x + 2)(2x – 1)(x – 3)。

6. Factoring Cubic Polynomials | 三次多项式的因式分解

A common application of the Factor Theorem is to factorise cubic polynomials. The standard approach is to find one root by trial, apply synthetic division, and then factor the resulting quadratic.

因式定理的一个常见应用是分解三次多项式。标准方法是试出一个根，应用综合除法，然后分解得到的二次式。

Example: Factorise x³ – 6x² + 11x – 6. Test x = 1: f(1) = 1 – 6 + 11 – 6 = 0, so (x – 1) is a factor. Dividing gives x² – 5x + 6, which factors as (x – 2)(x – 3). Therefore x³ – 6x² + 11x – 6 = (x – 1)(x – 2)(x – 3).

示例：分解 x³ – 6x² + 11x – 6。试 x=1：f(1) = 1 – 6 + 11 – 6 = 0，因此 (x-1) 是因式。除以 (x-1) 得 x² – 5x + 6，它可分解为 (x-2)(x-3)。所以 x³ – 6x² + 11x – 6 = (x – 1)(x – 2)(x – 3)。

x³ – 6x² + 11x – 6 = (x – 1)(x – 2)(x – 3)

Notice that each factor corresponds to a root of the original cubic. Conversely, if a cubic factors completely into three linear factors, then its three roots are immediately known.

注意每个因式都对应原三次方程的一个根。反过来，若一个三次多项式能完全分解为三个一次因式，则它的三个根立刻可知。

7. Solving Polynomial Equations | 解多项式方程

To solve a polynomial equation f(x) = 0, one can first use the Factor Theorem to find a linear factor, then continue factoring until the equation is reduced to linear and quadratic factors, and finally solve each factor equal to zero.

解多项式方程 f(x)=0 时，可先用因式定理找到一个一次因式，然后继续分解，直到方程化为一次和二次因式的乘积，最后令每个因式等于零求解。

Solve x³ – 2x² – 5x + 6 = 0. Testing x = 1 gives 1 – 2 – 5 + 6 = 0, so (x – 1) is a factor. Dividing yields x² – x – 6, which factors as (x – 3)(x + 2). Hence the equation becomes (x – 1)(x – 3)(x + 2) = 0, so x = 1, x = 3, or x = -2.

解方程 x³ – 2x² – 5x + 6 = 0。试 x=1 得 1 – 2 – 5 + 6 = 0，所以 (x-1) 是因式。除以 (x-1) 得 x² – x – 6，它分解为 (x-3)(x+2)。于是方程化为 (x-1)(x-3)(x+2) = 0，所以 x=1、x=3 或 x=-2。

A polynomial equation of degree n has at most n real roots. The Factor Theorem helps us find all of them systematically when the polynomial can be factored.

n 次多项式方程至多有 n 个实根。当多项式可以分解时，因式定理帮助我们系统地找到所有实根。

8. Repeated Roots and Multiplicity | 重根与重数

If f(a) = 0 and the factor (x – a) appears more than once in the factorisation, then a is a repeated root. The number of times (x – a) appears is called the multiplicity of the root.

若 f(a)=0 且因式 (x-a) 在分解式中出现不止一次，则 a 是重根。(x-a) 出现的次数称为该根的重数。

For example, f(x) = (x – 1)²(x + 2) has root x = 1 with multiplicity 2 and root x = -2 with multiplicity 1. We can detect a repeated factor by performing synthetic division twice or by checking whether f'(a) = 0 for differentiable polynomials.

例如，f(x) = (x – 1)²(x + 2) 的根 x=1 的重数为 2，根 x=-2 的重数为 1。我们可以通过连续两次综合除法，或对可导多项式检查 f'(a)=0 来发现重因式。

To test whether (x – a)² is a factor, first divide f(x) by (x – a); if the quotient also satisfies the Factor Theorem at x = a, then the factor appears at least twice.

要检验 (x-a)² 是否为因式，可先将 f(x) 除以 (x-a)；若商式在 x=a 处仍满足因式定理，则该因式至少出现两次。

9. Applications to Higher-Degree Polynomials | 在高次多项式中的应用

For polynomials of degree four or higher, the Factor Theorem can be applied repeatedly to reduce the degree step by step, eventually reaching a product of linear and quadratic factors.

对于四次或更高次的多项式，可以反复使用因式定理逐步降次，最终化为一次和二次因式的乘积。

Consider f(x) = x⁴ – 5x³ + 5x² + 5x – 6. Testing x = 1 gives 1 – 5 + 5 + 5 – 6 = 0, so (x – 1) is a factor. After division, the quotient is x³ – 4x² + x + 6. Testing x = 2 gives 8 – 16 + 2 + 6 = 0, so (x – 2) is a factor of the quotient. Dividing again gives x² – 2x – 3, which factors as (x – 3)(x + 1). Thus f(x) = (x – 1)(x – 2)(x – 3)(x + 1).

考虑 f(x) = x⁴ – 5x³ + 5x² + 5x – 6。试 x=1 得 1 – 5 + 5 + 5 – 6 = 0，因此 (x-1) 是因式。除以 (x-1) 得 x³ – 4x² + x + 6。试 x=2 得 8 – 16 + 2 + 6 = 0，所以 (x-2) 是该商式的因式。再次相除得 x² – 2x – 3，它分解为 (x-3)(x+1)。因此 f(x) = (x – 1)(x – 2)(x – 3)(x + 1)。

10. Common Pitfalls and Practice Tips | 常见错误与练习技巧

Do not confuse the root with the factor: if f(a) = 0, then the factor is (x – a), not (x + a).
不要混淆根与因式：若 f(a)=0，因式是 (x-a)，而不是 (x+a)。
Always test negative and fractional candidates, not just positive integers.
始终要检验负数和分数候选值，而不只是正整数。
After finding one factor, continue factoring the quotient completely; do not stop prematurely.
找到一个因式后，要继续将商式完全分解，不要过早停止。
Check for repeated factors by testing the quotient again with the same value of a.
通过对同一个 a 值再次检验商式，注意检查重因式。
Verify your final factorisation by expanding it back to the original polynomial.
最后将分解式展开，验证是否与原多项式一致。

11. Summary | 总结

The Factor Theorem is a powerful tool that connects roots and linear factors of polynomials. It is especially useful for factoring cubic and higher-degree polynomials, solving polynomial equations, and understanding the multiplicity of roots.

因式定理是联系多项式根与一次因式的有力工具。它特别适用于分解三次及更高次多项式、求解多项式方程以及理解根的重数。

By combining the Factor Theorem with synthetic division and the Rational Root Theorem, students can systematically solve many algebra problems that appear in examinations. Mastery of this topic is essential for progressing to topics such as partial fractions, curve sketching, and series.

将因式定理与综合除法、有理根定理相结合，学生可以系统地解决考试中出现的许多代数问题。掌握这一知识点，对进一步学习部分分式、曲线绘图和级数等内容至关重要。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Factor Theorem and Its Applications | 因式定理及其应用