A level revision

Binomials

A binomial is any expression that contains two terms, such as (1 + x)³ where the two terms are 1 and x respectively. Binomials like this can be expanded by multiplying out the brackets as with any polynomial. This is fine when the power is 2, 3 or 4 but with higher powers it becomes messy and time consuming. Another method is available: binomial expansion.

Binomial expansion

Firstly consider this expansion:

(a + b)³ = a³ + 3a²b + 3a² + b³

The numbers in front of each term (1, 3, 3 and 1 respectively) are the binomial coefficients and can be calcelated using 'Pascal's triangle':

Image courtesy of Wikipedia

Each number in the triangle is made from adding the two numbers directly above it, starting with a top row of 1, 1. It is important to note that the first number in each row is referred to as a '0th' term, rather than the 1st as you might expect!

In the earlier expansion example you can see that the coefficients of the terms can be found by just referring the 3rd row (the power) of Pascal's triangle.

Pascal's triangle still gets a bit complicated and time consuming past the first few rows however and so there is another method to calculate the coefficient of a term, referred to as ⁿC_r or it uses the following formula:

Where n is the power and r is the number of the term you are calculating (remember that r starts at 0). For example the coefficient of the second term of a binomial of power 3 is:

This matches the answer from Pascal's triangle and the expansion 'by hand'.

That is the coefficients out of the way, now what about the rest of term? The rules to remember are that the powers in each term always add together to give the power of the expansion, the first term of the expansion decreases in power from the power of the expansion and the second term increases to the power of the expansion. Ignoring the coefficients for the moment consider the following expansion:

(a + b)⁵ = a⁵ + a⁴b + a³b² + a²b³ + ab⁴ + b⁵

In each of the terms the powers add to give give 5: 5 + 0, 4 + 1, 3 + 2 and so on. The first term of the binomial, a, decreases in power from 5 to 0 and the second term, b, increases from 0 to 5. Putting this together with the coefficient formula and notation gives the binomial theorem:

Worked example

Expand fully (a + 2b)⁴

From Pascal's triangle the coefficients are 1, 4, 6, 4 and 1. You can then write the first line of our expansion:

(a + 2b)⁴ = 1 + 4 + 6 + 4 + 1

You can then add in the powers from the general binomial theorem:

(a + 2b)⁴ = a⁴ + 4a³b + 6a²2² + 4ab³ + b⁴

Now you substitute a and b in your expression for the a and b in the expansion you have been given, in this case a and 2b. Tip: Use brackets as it'll make it far easier when it comes to multiplying by the coefficients later.

(a + 2b)⁴ = (a)⁴ + 4(a)³(2b) + 6(a)²(2b)² + 4(a)(2b)³ + (2b)⁴

Now you should evaluate any powers, for example (2b)² can be written as 4b²:

(a + 2b)⁴ = (a)⁴ + 4(a)³(2b) + 6(a)²4b² + 4(a)8b³ + 16b⁴

Last step is to clean the expression up by multiplying by the coefficients, e.g. 6a4b² should become 24ab²

(a + 2b)⁴ = a⁴ + 8a³b + 24a²b² + 32ab³ + 16b⁴

Binomial series

The binomial series is the expansion of (1 + x)ⁿ where x is between -1 and 1. The binomial series can be calculated using the general case formula:

Worked example

Write down the first 4 terms of (1 + 3x)⁶

Firstly rewrite the series formula but replace n with 6 and x with 3x:

From here it's just a case of evaluating the expression. Starting with the fractions:

Now you can evaluate any powers:

Lastly multiple any numbers left in each term together...

Series as an approximation

You can use the binomial series for approximations such as the value of (0.99)⁹. To this simply rewrite the value in the form of (1 + x)ⁿ. In this case it would be (1 - x)⁹ with x = 0.01. Then it's a matter of working out the series to the required number of terms as above and substituting x with 0.01 and working out the value.