14.3 Binomial Theorem, Expansion Terms, and Coefficients

14.3 Binomial Theorem, Expansion Terms, and Coefficients

The Binomial Theorem establishes an algebraic identity bridging combinatorics, polynomial algebra, and numerical approximation. In Gaokao Mathematics, binomial theorem problems test structural manipulation of general terms, distinction between binomial coefficients and term coefficients, coefficient properties, and value assignment techniques.

1. Statement and Derivation of the Binomial Theorem

1.1 The Binomial Expansion Formula

For any positive integer $n \in \mathbb{N}^*$ and real numbers $a, b \in \mathbb{R}$:

$(a+b)^n = C_n^0 a^n + C_n^1 a^{n-1}b + C_n^2 a^{n-2}b^2 + \dots + C_n^r a^{n-r}b^r + \dots + C_n^n b^n = \sum_{r=0}^n C_n^r a^{n-r}b^r$

Combinatorial Derivation: Consider expanding $(a+b)^n = (a+b)(a+b)\cdots(a+b)$ ($n$ factors). Each term in the expanded polynomial is formed by choosing either $a$ or $b$ from each of the $n$ binomial factors. To obtain the term $a^{n-r}b^r$, we must choose $b$ from exactly $r$ factors and $a$ from the remaining $n-r$ factors. The number of ways to choose $r$ factors out of $n$ to supply $b$ is $C_n^r$. Thus, the coefficient of $a^{n-r}b^r$ is $C_n^r$.

1.2 The General Term Formula (通项公式)

The $(r+1)$-th term in the expansion of $(a+b)^n$ is designated as the general term $T_{r+1}$:

$T_{r+1} = C_n^r a^{n-r} b^r \quad (r = 0, 1, 2, \dots, n)$

[!IMPORTANT] Critical Distinction in Gaokao Questions:

1. Index Alignment: $T_{r+1}$ represents the $(r+1)$-th term, not the $r$-th term. For instance, the 4th term corresponds to $r = 3$.
2. Binomial Coefficient vs. Term Coefficient:
   • Binomial Coefficient (二项式系数): Refers strictly to the combinatorial number $C_n^r$. It depends solely on $n$ and $r$ and is always positive.
   • Term Coefficient (项的系数): Refers to the complete numerical multiplier of the algebraic variable in $T_{r+1}$, incorporating signs and constant coefficients present in $a$ and $b$.

2. Fundamental Properties and Coefficient Evaluation Strategies

2.1 Symmetry and Monotonicity of Binomial Coefficients

1. Symmetry: $C_n^r = C_n^{n-r}$. Terms equidistant from the two ends have equal binomial coefficients.
2. Maximum Binomial Coefficient (二项式系数的最大值):
   • If $n$ is even, the middle term $r = \frac{n}{2}$ has the unique maximum binomial coefficient $C_n^{n/2}$.
   • If $n$ is odd, the two central terms $r = \frac{n-1}{2}$ and $r = \frac{n+1}{2}$ share the maximum binomial coefficient $C_n^{(n-1)/2} = C_n^{(n+1)/2}$.

2.2 Sum Identities of Binomial Coefficients

By evaluating $(1+x)^n = \sum_{r=0}^n C_n^r x^r$ at specific values of $x$:

1. Total Sum of Binomial Coefficients: Set $x = 1 \implies C_n^0 + C_n^1 + C_n^2 + \dots + C_n^n = 2^n$
2. Alternating Sum: Set $x = -1 \implies C_n^0 - C_n^1 + C_n^2 - C_n^3 + \dots + (-1)^n C_n^n = 0$
3. Sums of Even and Odd Power Coefficients: Adding and subtracting the two results yields: $C_n^0 + C_n^2 + C_n^4 + \dots = C_n^1 + C_n^3 + C_n^5 + \dots = 2^{n-1}$

2.3 The Value Assignment Method (赋值法)

For a polynomial expansion $f(x) = (a_0 + a_1 x + a_2 x^2 + \dots + a_m x^m)^n = A_0 + A_1 x + A_2 x^2 + \dots + A_{mn} x^{mn}$:

• Constant Term $A_0$: Evaluate $f(0)$.
• Sum of All Coefficients $\sum A_k$: Evaluate $f(1)$.
• Alternating Coefficient Sum $\sum (-1)^k A_k$: Evaluate $f(-1)$.
• Sum of Even-Indexed Coefficients ($A_0 + A_2 + A_4 + \dots$): Calculate $\frac{f(1) + f(-1)}{2}$.
• Sum of Odd-Indexed Coefficients ($A_1 + A_3 + A_5 + \dots$): Calculate $\frac{f(1) - f(-1)}{2}$.

3. Classic Gaokao Worked Examples and Solutions

Worked Example 1: Finding Specific Coefficients in Rational Power Expansions

Problem Statement: In the expansion of $\left(2x^2 - \frac{1}{\sqrt{x}}\right)^9$:

1. Find the term containing $x^3$.
2. Determine the binomial coefficient and the term coefficient of $x^3$.

Detailed Mathematical Solution:

• Step 1: Write the general term $T_{r+1}$. Here $n = 9$, $a = 2x^2$, and $b = -x^{-1/2}$. $T_{r+1} = C_9^r (2x^2)^{9-r} \left(-x^{-1/2}\right)^r$
• Step 2: Simplify powers of $x$ and constants. $T_{r+1} = C_9^r \cdot 2^{9-r} \cdot (x^2)^{9-r} \cdot (-1)^r \cdot x^{-r/2}$ $T_{r+1} = (-1)^r \cdot 2^{9-r} \cdot C_9^r \cdot x^{2(9-r) - \frac{r}{2}} = (-1)^r \cdot 2^{9-r} \cdot C_9^r \cdot x^{18 - \frac{5r}{2}}$
• Step 3: Set the exponent of $x$ equal to $3$. $18 - \frac{5r}{2} = 3 \implies \frac{5r}{2} = 15 \implies 5r = 30 \implies r = 6$ Since $r = 6 \in {0, 1, \dots, 9}$, the term exists and is $T_7$ (the 7th term).
• Step 4: Compute Binomial Coefficient and Term Coefficient.
  • Binomial Coefficient: $C_9^6 = C_9^3 = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$
  • Term Coefficient: $\text{Coefficient} = (-1)^6 \cdot 2^{9-6} \cdot C_9^6 = 1 \cdot 2^3 \cdot 84 = 8 \times 84 = 672$
  • The term is $672 x^3$.

Worked Example 2: Product Expansion Coefficient Calculation

Problem Statement: Find the coefficient of $x^4$ in the expansion of $(1 + 2x)^5 (1 - x)^4$.

Detailed Mathematical Solution:

• Step 1: Express individual expansions.

  • Expansion of $(1+2x)^5$: General term $A_{r+1} = C_5^r (2x)^r = C_5^r \cdot 2^r \cdot x^r$ for $r \in {0, 1, 2, 3, 4, 5}$.
  • Expansion of $(1-x)^4$: General term $B_{k+1} = C_4^k (-x)^k = C_4^k (-1)^k x^k$ for $k \in {0, 1, 2, 3, 4}$.

• Step 2: Form the combined power $x^4$. The product of a term with $x^r$ and a term with $x^k$ gives $x^{r+k}$. We require $r + k = 4$ where $0 \le r \le 5$ and $0 \le k \le 4$. We list all non-negative integer pairs $(r, k)$:

  1. Pair $(4, 0)$: $r = 4, k = 0$
  2. Pair $(3, 1)$: $r = 3, k = 1$
  3. Pair $(2, 2)$: $r = 2, k = 2$
  4. Pair $(1, 3)$: $r = 1, k = 3$
  5. Pair $(0, 4)$: $r = 0, k = 4$

• Step 3: Calculate product terms for each pair.

  • Pair $(4, 0)$: $\left[C_5^4 2^4\right] \times \left[C_4^0 (-1)^0\right] = (5 \times 16) \times 1 = 80$
  • Pair $(3, 1)$: $\left[C_5^3 2^3\right] \times \left[C_4^1 (-1)^1\right] = (10 \times 8) \times (4 \times -1) = 80 \times (-4) = -320$
  • Pair $(2, 2)$: $\left[C_5^2 2^2\right] \times \left[C_4^2 (-1)^2\right] = (10 \times 4) \times (6 \times 1) = 40 \times 6 = 240$
  • Pair $(1, 3)$: $\left[C_5^1 2^1\right] \times \left[C_4^3 (-1)^3\right] = (5 \times 2) \times (4 \times -1) = 10 \times (-4) = -40$
  • Pair $(0, 4)$: $\left[C_5^0 2^0\right] \times \left[C_4^4 (-1)^4\right] = 1 \times 1 = 1$

• Step 4: Sum all partial coefficients. $\text{Total Coefficient of } x^4 = 80 - 320 + 240 - 40 + 1 = -39$