The binomial theorem gives a fast, organized way to expand powers such as (a + b)^n without multiplying the binomial by itself many times. It matters because these expansions appear in algebra, probability, combinatorics, and calculus. The theorem reveals that the coefficients are not random, but follow a precise counting pattern.
This pattern is the same one shown in Pascal’s Triangle.
Key Facts
- Binomial theorem: (a + b)^n = sum from k = 0 to n of C(n,k)a^(n-k)b^k.
- Binomial coefficient: C(n,k) = n! / (k!(n-k)!).
- The powers of a decrease from n to 0, while the powers of b increase from 0 to n.
- The coefficients in (a + b)^n are the numbers in row n of Pascal’s Triangle, counting the top row as row 0.
- Pascal’s rule: C(n,k) = C(n-1,k-1) + C(n-1,k).
- Example: (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
Vocabulary
- Binomial
- A binomial is an algebraic expression with exactly two terms, such as a + b or x - 3.
- Binomial theorem
- The binomial theorem is a formula for expanding (a + b)^n into a sum of terms with binomial coefficients.
- Binomial coefficient
- A binomial coefficient C(n,k) counts how many ways to choose k objects from n objects.
- Pascal’s Triangle
- Pascal’s Triangle is a triangular number pattern where each interior number is the sum of the two numbers above it.
- Exponent
- An exponent tells how many times a base is used as a factor.
Common Mistakes to Avoid
- Writing (a + b)^n as a^n + b^n is wrong because powers do not distribute over addition.
- Using the wrong row of Pascal’s Triangle is wrong because (a + b)^n uses row n when the top row is counted as row 0.
- Forgetting that the exponents in each term add to n is wrong because every term comes from multiplying n total factors.
- Dropping the coefficient 1 at the ends is wrong because the first and last terms still have coefficients C(n,0) = 1 and C(n,n) = 1.
Practice Questions
- 1 Expand (x + 2)^4 completely.
- 2 Find the coefficient of x^3y^2 in the expansion of (x + y)^5.
- 3 Explain why the coefficients in the expansion of (a + b)^6 match a row of Pascal’s Triangle.