Binomial Theorem Formulas
IB Mathematics Analysis & Approaches
🎯 The Binomial Theorem
General Formula:
The binomial theorem provides a formula to expand \((a + b)^n\) where \(n\) is a positive integer.
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
where \(n \in \mathbb{N}\) (positive integer) and \(\binom{n}{k}\) is the binomial coefficient
Expanded Form:
\[(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n\]
The expansion contains \(n+1\) terms
📊 Binomial Coefficient
Formula:
The binomial coefficient \(\binom{n}{k}\) (read as "n choose k") represents the number of ways to choose \(k\) elements from \(n\) elements.
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
where \(0 \leq k \leq n\) and \(n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1\)
Calculator Notation:
\(\binom{n}{k}\) is often written as \(^nC_k\) or \(C(n,k)\) on calculators
Special Values:
\[\binom{n}{0} = 1\]
\[\binom{n}{1} = n\]
\[\binom{n}{n} = 1\]
🔢 General Term (Finding a Specific Term)
The (k+1)th Term:
To find a specific term in the expansion without expanding fully, use the general term formula.
\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\]
where \(T_{k+1}\) is the \((k+1)\)th term and \(k\) starts from 0
Finding the Coefficient of a Specific Term:
To find the coefficient of a term with a specific power of \(x\):
- Identify the general term formula for your expansion
- Set the power of the variable equal to the desired power
- Solve for \(k\)
- Substitute \(k\) back into the general term to find the coefficient
🔺 Pascal's Triangle
Structure:
Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. Each row gives the binomial coefficients for \((a+b)^n\).
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Relationship to Binomial Coefficients:
Row \(n\) of Pascal's Triangle gives the coefficients for \((a+b)^n\)
For example: Row 4 gives 1, 4, 6, 4, 1, so \((a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\)
⚡ Important Properties
Symmetry Property:
\[\binom{n}{k} = \binom{n}{n-k}\]
The coefficients are symmetric across the expansion
Number of Terms:
\((a+b)^n\) has \(n+1\) terms
Power Pattern:
Powers of \(a\) decrease from \(n\) to 0
Powers of \(b\) increase from 0 to \(n\)
The sum of powers in each term equals \(n\)
Sum of Coefficients:
\[\sum_{k=0}^{n} \binom{n}{k} = 2^n\]
This can be found by substituting \(a=1\) and \(b=1\) into \((a+b)^n\)
🌟 Special Cases
Square of a Binomial:
\[(a + b)^2 = a^2 + 2ab + b^2\]
Cube of a Binomial:
\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
Binomial with Negative Second Term:
\[(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k\]
Signs alternate: positive for even powers of \(b\), negative for odd powers
📝 Quick Reference Examples
Example 1: \((x+1)^4\)
\[(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1\]
Example 2: \((2x+3)^3\)
\[(2x+3)^3 = 8x^3 + 36x^2 + 54x + 27\]
Example 3: \((x-2)^3\)
\[(x-2)^3 = x^3 - 6x^2 + 12x - 8\]
💡 Pro Tip: Use Pascal's Triangle for quick expansions with small powers, and use the general term formula when finding specific terms or coefficients in larger expansions.