Binomial Theorem

I think that on first seeing the Binomial theorem many students think it very complicated and straight away tell themselves that this is too difficult to deal with. But like many other things in maths if we just identify a few patterns in how things are ordered it suddenly appears much simpler.

It is binomial because we are working with the sum or difference of two algebraic terms, (x+y). It tells us the result of raising this to any whole number power. Lets take a few examples that we know already or can easily work out by multiplying out the (x+y) the required number of times. Remember from the rules of indices that anything to the power of zero is 1.

                (x+y)⁰ = 1                                       1 term

                (x+y)¹ = x+y                                   2 terms

                (x+y)² = x² +2xy + y²                                  3 terms

                (x+y)³ = x³ + 3x²y + 3xy² + y³             4 terms

If we raise it to the power of n, then the number of terms in the expansion will be n + 1, and these are all added together.

Look at the powers of x. They start at the same power as the binomial is being raised to and then decrease by 1 in each succeeding term until we reach x⁰. Remember x⁰ is 1. And since multiplying by 1 makes no difference we do not need to write it in.

The powers of y follow the reverse pattern starting at y⁰ in the first term. Again since this is 1 and it is multiplied by xⁿ it is just ignored. They increase by 1 in each succeeding term until we reach yⁿ.

We then have to find the coefficients of the different terms. These are given by calculating ⁿC_r where n is the power we are raising the binomial to, and r starts at 0 for the first term and increments by 1 in each succeeding term until it reaches n.

The booklet gives the definition of ⁿC_r in terms of factorials but it is much simpler to just use the nCr button on your calculator.

In general then a binomial expansion raised to the power of n will consist of the sum of n+1 terms of the form ⁿC_r . x^n-r .y^r where r starts at zero and increases by 1 in succeeding terms.

This results in

General term of binomial expansion

= T_r+1 = ⁿC_r . x^n-r .y^r

Find the term independent of x in (x² – 1/x)¹⁵

The term independent of x is the term with no x, in other words the index (power) on x is zero.

T_r+1 = ¹⁵C_r . (x²)^15-r .(-1/x)^r

= ¹⁵C_r . (x)^30-2r .(-1/x)^r

= ¹⁵C_r . x^30-2r .(-x)-^r

= ¹⁵C_r . -x^30-3r

30-3r = 0 for independent term. r = 10

T_r+1 = ¹⁵C_r . (x²)^15-r .(-1/x)^r = 3003 . x¹⁰ . 1/(-x¹⁰) = 3003