The Binomial Theorem

This chapter is about the expansion of (x + y)^n, where n is a positive integer (or zero). When you have completed it, you should

1. be able to use Pascal's triangle to find the expansion of (x + y)^n when n is small.
2. know how to calculate the coefficients in the expansion of (x + y)^n when n is large
3. be able to use notation (n r) *r is below n* in the context of the binomial theorem.

1. Expanding (x + y)^n
The binomial expansion is about calculating (x +y)^n quickly and easily. It is useful to start by looking at (x + y )^n for n = 2,3 and 4.

The expansions are
:

You can summarize these results, including (x + y)^1, as follows. The coefficient are in bold type.

Study these expansions carefully. Notice how the powers start from the left with x^n. The powers of x then successively reduce by 1, and the powers of y increase by 1 until reaching the term y^n.
Notice also that the coefficient from the pattern of Pascal's triangle.

Ex. Write down the expansion of (1 + y)^6
Use the next row of Pascal's triangle, continuing the pattern of powers and replacing x by 1 :

Ex. Multiply out the brackets in the expression (2x + 3)^4.
Use the expansion of (x + y)^4, replacing x by (2x) and replacing y by 3 :

Ex. Expand (x2 + 2)^3.

Ex. Find the coefficient of x3 in the expansion of (3x - 4)^5.

Ex. Expand (1 + 2x +3x2)^3

2. The binomial theorem

The treatment giveN above is fine for finding the coefficients in the expansion of (x + y)^n where n is small, but it is hopelessly inefficient for finding the coefficient of x11y4 in the expansion of (x + y)^15. Just think of all the rows of pascal's triangle which you would have to write out !
What you need is a formula in terms of n and r for the coefficient of x^n-r y^r in the expansion (x + y)^n.

This formula will be given in the formula list. however, you should how how to substitute the values correctly.

Exercises with solutions will be uploaded as soon as possible.

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