11.6Binomial Theorem
Expand (a+b)ⁿ using Pascal's Triangle and the Binomial Theorem, and find any specific term without expanding the full expression.
The Binomial Theorem is a cornerstone of algebra, probability, and calculus. It appears in the binomial distribution, Newton's generalized binomial series, and combinatorial identities used throughout higher mathematics.
Essential Question
How can Pascal's Triangle and the Binomial Theorem help us efficiently expand powers of binomials?
Overview
The Binomial Theorem gives a formula for expanding any power of a binomial (a + b)ⁿ without multiplying it out step by step. The coefficients in each expansion come directly from Pascal's Triangle or from binomial coefficients C(n, k) = n! / [k!(n − k)!].
Binomial Theorem: (a + b)ⁿ = Σk=0n C(n, k) · aⁿ⁻ᵏ · bᵏ
The (k+1)th term of the expansion is: C(n, k) · aⁿ⁻ᵏ · bᵏ
Pascal's Triangle — Rows 0 through 5
Each entry = sum of the two entries above it. Row 4 (highlighted) gives coefficients for (a+b)⁴.
Binomial
A polynomial with exactly two terms, e.g. (a + b).
Pascal's Triangle
A triangular array where each entry is the sum of the two entries above it; rows give binomial coefficients.
Binomial Coefficient C(n,k)
The number of ways to choose k items from n: C(n,k) = n!/[k!(n−k)!]. Also written ⁿCₖ or (n choose k).
Binomial Theorem
(a+b)ⁿ = Σ(k=0 to n) C(n,k)·aⁿ⁻ᵏ·bᵏ — gives every term of the expansion.
General Term
The (k+1)th term of (a+b)ⁿ is C(n,k)·aⁿ⁻ᵏ·bᵏ.
Worked Examples
Expand (x + y)⁴ using the Binomial Theorem.
Identify: a = x, b = y, n = 4.
Write the sum: Σ(k=0 to 4) C(4,k)·x⁴⁻ᵏ·yᵏ.
k=0: C(4,0)·x⁴·y⁰ = 1·x⁴ = x⁴
k=1: C(4,1)·x³·y¹ = 4x³y
k=2: C(4,2)·x²·y² = 6x²y²
k=3: C(4,3)·x¹·y³ = 4xy³
k=4: C(4,4)·x⁰·y⁴ = y⁴
Combine all terms.
Expand (2x − 3)³.
Rewrite as (2x + (−3))³. So a = 2x, b = −3, n = 3.
k=0: C(3,0)·(2x)³·(−3)⁰ = 1·8x³·1 = 8x³
k=1: C(3,1)·(2x)²·(−3)¹ = 3·4x²·(−3) = −36x²
k=2: C(3,2)·(2x)¹·(−3)² = 3·2x·9 = 54x
k=3: C(3,3)·(2x)⁰·(−3)³ = 1·1·(−27) = −27
Combine all terms.
Find the 4th term of (x + 2)⁶.
The (k+1)th term formula: C(n,k)·aⁿ⁻ᵏ·bᵏ.
4th term → k+1 = 4 → k = 3.
n = 6, a = x, b = 2.
C(6,3) = 6!/(3!·3!) = 720/(6·6) = 20.
Term = C(6,3)·x⁶⁻³·2³ = 20·x³·8 = 160x³.
Find the coefficient of x³y² in (x + y)⁵.
Use the Binomial Theorem: (x+y)⁵ = Σ(k=0 to 5) C(5,k)·x⁵⁻ᵏ·yᵏ.
We need x³y², so 5−k = 3 → k = 2.
Coefficient = C(5,2) = 5!/(2!·3!) = 10.
Use Pascal's Triangle to expand (a + b)⁵.
Row 5 of Pascal's Triangle: 1, 5, 10, 10, 5, 1.
These are the coefficients for k = 0, 1, 2, 3, 4, 5.
k=0: 1·a⁵ = a⁵
k=1: 5·a⁴b
k=2: 10·a³b²
k=3: 10·a²b³
k=4: 5·ab⁴
k=5: 1·b⁵
Write out the full expansion.
Guided Practice
Expand (x + 1)⁴ using the Binomial Theorem.
Hint: Use a = x, b = 1, n = 4. Row 4 of Pascal's Triangle is 1, 4, 6, 4, 1.
Expand (x − 2)³.
Hint: Rewrite as (x + (−2))³. Apply the Binomial Theorem with a = x, b = −2, n = 3. Watch the signs!
Find the 3rd term of (x + 3)⁵.
Hint: 3rd term → k = 2. Use C(5,2)·x⁵⁻²·3².
Find the coefficient of x²y³ in (x + y)⁵.
Hint: Set 5 − k = 2, so k = 3. The coefficient is C(5, 3).
Write row 6 of Pascal's Triangle.
Hint: Start with 1. Each entry is the sum of the two entries directly above it in row 5: 1, 5, 10, 10, 5, 1.
Common Mistakes
Writing (a + b)³ = a³ + b³ (forgetting middle terms).
(a + b)³ = a³ + 3a²b + 3ab² + b³ — always use the Binomial Theorem or Pascal's Triangle.
Forgetting to raise b to the correct power when b is negative, e.g. (−3)² = −9.
(−3)² = +9. Negative numbers raised to even powers are positive.
Using k instead of k+1 when identifying the term number (off-by-one error).
The (k+1)th term corresponds to index k. For the 4th term, k = 3.
Computing C(n,k) as n!/k! instead of n!/[k!(n−k)!].
Always divide by both k! and (n−k)! in the binomial coefficient formula.
Math Tips
Pascal's Triangle is fastest for small n (n ≤ 6); use the formula C(n,k) for larger n.
The exponents of a decrease from n to 0 while the exponents of b increase from 0 to n — they always sum to n.
The expansion of (a+b)ⁿ has exactly n+1 terms.
The coefficients are symmetric: C(n,k) = C(n, n−k).
To find a specific term quickly, use the general term formula instead of expanding fully.
Interactive Practice — 1 Questions
Which row of Pascal's Triangle gives the coefficients for (a + b)⁴?