📚 IB Mathematics: Core Properties of Polynomials with Real Coefficients | IB数学：实系数多项式的核心性质
In IB Mathematics, polynomials with real coefficients form a foundational topic bridging algebra, functions, and calculus. Understanding their properties is essential for solving problems involving roots, factors, graphs, and inequalities.
在IB数学中，实系数多项式是连接代数、函数与微积分的基础主题。理解其性质对于解决涉及根、因式、图像和不等式的问题至关重要。
1. Definition and Standard Form | 定义与标准形式
A polynomial with real coefficients is an expression of the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where each aᵢ is a real number and n is a non-negative integer. The highest power n (with aₙ ≠ 0) determines the degree.
实系数多项式是形如 P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中每个 aᵢ 都是实数，n 是非负整数。最高次幂 n（且 aₙ ≠ 0）决定其次数。
For example, P(x) = 3x⁴ − 2x² + 5x − 1 is a degree-4 polynomial with real coefficients. The domain is all real numbers, and the range depends on the degree and leading coefficient.
例如，P(x) = 3x⁴ − 2x² + 5x − 1 是四次实系数多项式。其定义域为全体实数，值域取决于次数和首项系数。
2. Degree and Leading Coefficient | 次数与首项系数
The degree of a polynomial is the highest exponent of x with a non-zero coefficient. The leading coefficient is the coefficient of that term. These two values determine the end behavior of the graph.
多项式的次数是 x 的最高非零指数。首项系数是该最高次项的系数。这两个值决定图像的两端趋势。
If deg P = n and leading coefficient aₙ > 0, then as x → +∞, P(x) → +∞; if aₙ < 0, then P(x) → −∞.
若 deg P = n 且首项系数 aₙ > 0，则当 x → +∞ 时，P(x) → +∞；若 aₙ < 0，则 P(x) → −∞。
For even-degree polynomials, both ends go in the same direction; for odd-degree polynomials, they go in opposite directions.
对于偶次多项式，两端趋势相同；对于奇次多项式，两端趋势相反。
3. Factor Theorem and Remainder Theorem | 因式定理与余数定理
The Remainder Theorem states that when a polynomial P(x) is divided by (x − a), the remainder is P(a). If P(a) = 0, then (x − a) is a factor, which is the Factor Theorem.
余数定理指出：多项式 P(x) 除以 (x − a) 所得的余数为 P(a)。若 P(a) = 0，则 (x − a) 是一个因式，这就是因式定理。
For example, for P(x) = x³ − 4x² + x + 6, P(2) = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor. Dividing gives P(x) = (x − 2)(x² − 2x − 3).
例如，对于 P(x) = x³ − 4x² + x + 6，P(2) = 8 − 16 + 2 + 6 = 0，因此 (x − 2) 是一个因式。相除得 P(x) = (x − 2)(x² − 2x − 3)。
4. Real Coefficients and Complex Roots | 实系数与复根
When a polynomial has real coefficients, any complex roots must occur in pairs. If a + bi is a root (with b ≠ 0), then its conjugate a − bi is also a root.
当多项式具有实系数时，任何复根必定成对出现。若 a + bi 是根（b ≠ 0），则其共轭 a − bi 也是根。
This property ensures that the polynomial can be factored into real quadratic factors corresponding to each complex conjugate pair.
这一性质确保多项式可以分解为对应每对共轭复根的实系数二次因式。
5. Conjugate Root Theorem in Action | 共轭复根定理的应用
Suppose P(x) is a real-coefficient polynomial of degree 3, and 2 + i is a root. Then 2 − i must also be a root. The quadratic factor is (x − (2 + i))(x − (2 − i)) = x² − 4x + 5.
假设 P(x) 是三次实系数多项式，且 2 + i 是一个根，则 2 − i 也必定是根。对应二次因式为 (x − (2 + i))(x − (2 − i)) = x² − 4x + 5。
Since the degree is 3, there must be a third real root, say r. Thus P(x) = a(x² − 4x + 5)(x − r), where a is a real constant.
由于次数为3，必定存在第三个实根 r。因此 P(x) = a(x² − 4x + 5)(x − r)，其中 a 为实常数。
6. Fundamental Theorem of Algebra | 代数基本定理
Every non-constant polynomial of degree n over the complex numbers has exactly n complex roots, counted with multiplicity. This includes real roots as special cases of complex roots (where the imaginary part is zero).
在复数范围内，每个 n 次非零多项式恰好有 n 个复根（按重数计）。实根是虚部为零的复根特例。
For real-coefficient polynomials, the non-real roots come in conjugate pairs, so the number of real roots has the same parity as the degree.
对于实系数多项式，非实根成对出现，因此实根的个数与次数具有相同的奇偶性。
7. Vieta’s Formulas | 韦达定理
Vieta’s formulas relate the roots to the coefficients. For a cubic P(x) = ax³ + bx² + cx + d with roots r₁, r₂, r₃:
韦达定理将根与系数联系起来。对于三次多项式 P(x) = ax³ + bx² + cx + d，根为 r₁, r₂, r₃：
r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = −d/a
r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = −d/a
For a quadratic ax² + bx + c with roots r₁ and r₂: sum = −b/a and product = c/a.
对于二次多项式 ax² + bx + c，根为 r₁ 和 r₂：和 = −b/a，积 = c/a。
8. Rational Root Theorem | 有理根定理
If a polynomial with integer coefficients has a rational root p/q in lowest terms, then p must divide the constant term and q must divide the leading coefficient.
如果整数系数多项式存在最简有理根 p/q，则 p 必须整除常数项，q 必须整除首项系数。
For P(x) = 2x³ − 3x² − 8x + 12, possible rational roots are ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2. Testing shows x = 2 is a root.
对于 P(x) = 2x³ − 3x² − 8x + 12，可能的有理根为 ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2。检验得 x = 2 是根。
9. Factoring Strategies | 因式分解策略
Start by finding one rational root via the Rational Root Theorem, then use synthetic division to reduce the polynomial. Continue factoring the quotient until a quadratic remains, which can be solved by the quadratic formula.
首先用有理根定理找到一个有理根，然后用综合除法降低多项式次数。继续分解商式直到剩下二次式，用求根公式解决。
For cubic polynomials, one real root always exists. Once found, the remaining quadratic may have two real roots or a conjugate complex pair.
三次多项式一定存在一个实根。找到后，剩余二次式可能有两条实根或一对共轭复根。
10. Graphs and Multiplicity of Roots | 图像与根的重数
If a root has odd multiplicity, the graph crosses the x-axis at that point. If the multiplicity is even, the graph touches the x-axis and turns around.
若根的重数为奇数，图像在该点穿过 x 轴；若重数为偶数，图像在该点接触 x 轴并折返。
For example, in P(x) = (x − 1)³(x + 2)², x = 1 has multiplicity 3 (cross), and x = −2 has multiplicity 2 (touch).
例如，在 P(x) = (x − 1)³(x + 2)² 中，x = 1 的重数为3（穿过），x = −2 的重数为2（接触）。
11. Polynomial Inequalities | 多项式不等式
To solve P(x) > 0 or P(x) < 0, first factor the polynomial completely and identify all real roots. These roots divide the number line into intervals. Test a point in each interval to determine the sign.
要解 P(x) > 0 或 P(x) < 0，首先将多项式完全分解并找出所有实根。这些根将数轴分成若干区间。在每个区间内取测试点判断符号。
Remember to exclude roots where the inequality is strict (> or
注意：对于严格不等式（> 或
12. Applications and Exam Tips | 应用与考试技巧
Real-coefficient polynomials model many real-world phenomena, from projectile motion to economic profit. In IB exams, common questions involve finding unknowns using given roots, sketching graphs, and solving inequalities.
实系数多项式可模拟许多现实现象，从抛体运动到经济利润。在IB考试中，常见题型包括利用已知根求未知系数、画图以及解不等式。
Always check the degree to determine the total number of roots, and use the conjugate root theorem immediately when a complex root is given. Practice synthetic division to save time.
始终通过次数确定根的总数，当给出复根时立即使用共轭根定理。练习综合除法以节省时间。
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