Binomial Theorem Introduction
Learn how to expand (a + b)n using binomial coefficients and Pascal's triangle.
The Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)n without having to multiply repeatedly. For any positive integer n:
(a + b)n = nC0 an + nC1 an-1b + nC2 an-2b2 + ... + nCn bn
Or using sigma notation: (a + b)n = Σk=0n nCk an-k bk
Each term in the expansion has the form nCk an-k bk, where k runs from 0 to n. Notice that:
- ●The powers of a decrease from n to 0
- ●The powers of b increase from 0 to n
- ●In each term, the sum of the powers of a and b equals n
- ●There are n + 1 terms in the expansion
Binomial Coefficients and Pascal's Triangle
The coefficients nCk in the expansion are called binomial coefficients. They can be read directly from Pascal's triangle.
Small Expansions Using Pascal's Triangle
Row 1: 1, 1
(a + b)1 = a + b
Row 2: 1, 2, 1
(a + b)2 = a2 + 2ab + b2
Row 3: 1, 3, 3, 1
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Row 4: 1, 4, 6, 4, 1
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
Finding Specific Terms
You do not always need to write the full expansion. The general term (the (k+1)th term) of (a + b)n is:
Tk+1 = nCk an-k bk
This is useful when asked to find a particular term or coefficient without fully expanding. For instance, to find the coefficient of x3 in (2 + x)5, identify which value of k gives x3 (here k = 3), then compute that single term.
Key Vocabulary
Binomial
An algebraic expression containing two terms, such as (a + b).
Binomial Coefficient
The number nCk that multiplies each term in the binomial expansion.
General Term
The formula Tk+1 = nCk an-k bk that gives any term in the expansion.
Expansion
The process of writing a power of a binomial as a sum of individual terms.
Worked Examples
Expand (x + 2)3.
Step 1: Use Pascal's row 3 coefficients: 1, 3, 3, 1. Here a = x, b = 2.
Step 2: = 1(x3) + 3(x2)(2) + 3(x)(22) + 1(23)
Step 3: = x3 + 6x2 + 12x + 8
Answer: (x + 2)3 = x3 + 6x2 + 12x + 8
Find the coefficient of x3 in the expansion of (1 + x)5.
Step 1: The general term is 5Ck(1)5-k(x)k = 5Ck xk.
Step 2: For x3, set k = 3: coefficient = 5C3 = 10.
Answer: The coefficient of x3 is 10.
Find the 4th term in the expansion of (2x - 1)5.
Step 1: The 4th term has k = 3. Here a = 2x, b = -1, n = 5.
Step 2: T4 = 5C3(2x)2(-1)3 = 10 × 4x2 × (-1)
Step 3: = 10 × (-4x2) = -40x2
Answer: The 4th term is -40x2.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
How many terms are in the expansion of (a + b)6?
Question 2
What is the coefficient of x2 in the expansion of (1 + x)4?
Question 3
Expand (a + b)2 using the binomial theorem.
Question 4
What is the constant term in the expansion of (3 + x)4?
Question 5
What is the coefficient of x2 in the expansion of (3 + x)5?
Key Concepts Summary
- ●The binomial theorem expands (a + b)n into a sum of n + 1 terms.
- ●Each term uses a binomial coefficient nCk from Pascal's triangle.
- ●The general term is Tk+1 = nCk an-k bk.
- ●Powers of a decrease while powers of b increase across the expansion.
- ●You can find a specific term without fully expanding by using the general term formula.