Binomial Theorem
The binomial theorem provides a formula for expanding powers of binomials (a + b)^n using Pascal's triangle and combination notation.
What You Need to Know
Key Concept Diagram
Pascal's triangle gives the coefficients for binomial expansions
The binomial theorem states (a+b)^n = sum of C(n,k) a^(n-k) b^k for k from 0 to n
C(n,k) = n! / (k!(n-k)!) is the binomial coefficient (n choose k)
The expansion of (a+b)^n has (n+1) terms
The general term is T(k+1) = C(n,k) a^(n-k) b^k
Key Vocabulary
Binomial
An algebraic expression with exactly two terms
Pascal's triangle
A triangular array of numbers where each number is the sum of the two above it
Binomial coefficient
The coefficient C(n,k) representing the number of ways to choose k items from n
General term
A formula that gives any specific term in a binomial expansion
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the coefficient of x^2 in the expansion of (1 + x)^4?
Question 2
Expand (x + 1)^3.
Question 3
How many terms are in the expansion of (a + b)^7?
Key Concepts Summary
- ●Pascal's triangle gives the coefficients for binomial expansions
- ●The binomial theorem states (a+b)^n = sum of C(n,k) a^(n-k) b^k for k from 0 to n
- ●C(n,k) = n! / (k!(n-k)!) is the binomial coefficient (n choose k)
- ●The expansion of (a+b)^n has (n+1) terms
- ●The general term is T(k+1) = C(n,k) a^(n-k) b^k