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.
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.
