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OCR A-level Maths Binomial expansion

Binomial expansion

This page covers the following topics:

1. Factorials
2. Pascal's triangle
3. Binomial expansion

A factorial is the product of all whole numbers smaller than or equal to a given number. The symbol of the factorial function is !. Factorials can be calculated on a calculator by pressing the number followed by the factorial button (!).

Factorials

Pascal's triangle is a triangular pattern of numbers, beginning with a 1 on top, that are arranged so that each number is the sum of the two numbers directly above it. It can be used to determine the coefficients of a binomial expansion. When expanding a binomial of the type (x+y)n, the coefficients correspond to the numbers in the nth row of Pascal?s triangle (counting the top row, consisting of a 1, as the 0th row).

Pascal's triangle

For brackets with larger powers, it can be impractical to use Pascal's Triangle to determine the coefficients. Instead, they can be calculated by the formula NC(r) (see image). There is a button on the calculator for this. If n is a positive integer, the bracket can then be expanded using the binomial theorem: (a + b)n = an + (nC1)anβˆ’1b + (nC2)anβˆ’2b2 + … + (nCnβˆ’1)abnβˆ’1 + bn.

Binomial expansion

1

Find the coefficient of the second term of the expansion of (5x + 5)Β².

Find the coefficient of the second term of the expansion of (5x + 5)Β².

2

Find the first three terms of (4x + 5)⁢, in descending powers of x.

Find the first three terms of (4x + 5)⁢, in descending powers of x.

3

What is the symbol for factorial notation?

What is the symbol for factorial notation?

4

Calculate 9!.

Calculate 9!.

5

Find the coefficients of the expansion of (2x βˆ’ 8)Β³.

Find the coefficients of the expansion of (2x βˆ’ 8)Β³.

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