Understanding Pascal's Triangle for Binomial Expansions
Pascal's triangle is a triangular array of numbers that helps us find coefficients in binomial expansions. Each row corresponds to the coefficients of the expanded form of a binomial expression, such as (a + b)^n.
How to Build Pascal's Triangle:
- Start with a top row of 1.
- Each subsequent row is formed by adding the two numbers directly above it.
- The edges of the triangle are always 1.
For example, the first five rows of Pascal's triangle are:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
To find the coefficients for the expansion of (a + b)^4, look at the fifth row: 1, 4, 6, 4, 1. Thus, (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
Key Points:
- Each row corresponds to the power of the binomial.
- Coefficients are the sums of the two numbers above in the triangle.
- The nth row gives coefficients for (a + b)^n.
Key points to remember
- Pascal's triangle starts with 1 at the top.
- Each number is the sum of the two above it.
- Coefficients correspond to the nth row for (a + b)^n.
Worked example
Determine the coefficients for (x + y)^3 using Pascal's triangle.
Model Answer: The fourth row of Pascal's triangle is 1, 3, 3, 1. Thus, (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3.