BestMaths Online

The Binomial Theorem

An expression such as (2x + 3) is called a binomial. Expanding an expression such as (2x + 3)⁷ is called a binomial expansion. The main aim of this topic is to develop a method for such an expansion without having to multiply it out.

Before developing the Binomial Theorem a few other concepts need to be explored.

Factorials, Permutations and Combinations

Factorial

The factorial of a number n, is the product of all of the whole numbers up to and including n.

n! = n.(n − 1).(n − 2).(n − 3) ... 4.3.2.1        where n is a natural number

Permutations

A permutation is an arrangement of elements from a set where the order of the elements is considered.

e.g. There are 6 possible permutations of two elements from the set {a, b, c}. These are (ab, ac, ba, bc, ca, cb)

The formula to work out the number of permuations of r elements from a set of n elements written  is:

Example

How many permutations are there of three of the letters from the set {a, b, c, d, e}?

Combinations

A combination is an arrangement of elements from a set, where the order of the elements chosen is not considered .

e.g. The are only 3 possible combinations of two elements from the set {a, b, c}. These are (ab, bc, ac)

The formula to work out the number of combinations of r elements from a set of n elements written is:

The notation  can also be written .

Example

How many combinations are there of three of the letters from the set {a, b, c, d, e}?

The Binomial Coefficients

A coefficient is the number in an algebraic term. e.g. In 7x³ the coefficient is 7.

A binomial coefficient is the coefficient of a term in a binomial expansion of the form (a + b)ⁿ

Investigation of the binomial expansions for different values of n reveals some interesting results:

binomial	binomial expansion	binomial coefficients
(a + b)0	1	1
(a + b)1	a + b	1   1
(a + b)2	a2 + 2ab + b2	1   2   1
(a + b)3	a3 + 3a2b + 3ab2 + b3	1   3   3   1
(a + b)4	a4 + 4a3b + 6a2b2 + 4ab3 + b4	1   4   6   4   1
(a + b)5	a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4+ b5	1   5   10   10   5   1

Note

• The triangle of binomial coefficients is known as Pascal's Triangle.
• Note that each value is the sum of the two above it.
• Notice also that there is always (n + 1) terms for a binomial to the n th power.
• The exponent of a decreases by 1 from left to right.
• The exponent of b increases by 1 from left to right.

The binomial coefficients can also be found from the combination formula:

For (a + b)4 the coefficients are 1, 4, 6, 4, 1 which is the same as  . 
These combinations can be looked up in the table of Binomial Coefficients often provided for examinations. They can also be found on calculators.

All of this, leads to the Binomial Theorem

The Binomial Theorem

This states that:

It is true for all values of a and b and at this level we use it only for n being a positive integer. {1, 2, 3...}

Example

                             = 1.32x⁵ + 5.16.3.x⁴ + 10.8.9.x^{3 +} 10.4.27.x² + 5.2.81. x + 1.243

                             = 32x⁵ + 240x⁴ + 720x^{3 +} 1080x² + 810x + 243

The General Term

Often, especially when n is large a specific term is required. e.g the seventh term.

To do this the general term is used:

Note that this is the (r + 1)th term not the r th term

Example

Find the 7th term of (x + 3)¹²

Much easier than expanding out the bracket 12 times!

Approximations using the Binomial Theorem

It is possible to find approximate values for calculations such as 1.06⁴ (to 5 significant figures),using the binomial theorem

1.06⁴    = (1 + 0.06)⁴

             = 1.1⁴ + 4.1³ (0.06) + 6.1² (0.06)² + 4.1¹.(0.06)³ + 1.(0.06)⁴

              = 1 + 0.24 + 0.0216 + 0.000864 + 0.00001296

              = 1.26247696

              = 1.2625 (to 5 sig.fig.)