# 4. The Binomial Theorem

by M. Bourne

A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.

We sometimes need to expand binomials as follows:

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.

## Pascal's Triangle

We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

You can use this pattern to form the coefficients, rather than multiply everything out as we did above.

## The Binomial Theorem

We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.

## Properties of the Binomial Expansion (a + b)n

• There are n + 1 terms.
• The first term is an and the final term is bn.
• Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n.
• If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.

## General formula for (a + b)n

First, we need the following definition:

Definition: n! represents the product of the first n positive integers i.e.

n! = n(n − 1)(n − 2) ... (3)(2)(1)

We say n! as "n factorial".

### Example 1 - factorial values

Here are some factorial values:

(a) 3! = (3)(2)(1) = 6

(b) 5! = (5)(4)(3)(2)(1) = 120

(c) (4!)/(2!)=((4)(3)(2)(1))/((2)(1))=12

Note: (4!)/(2!) cannot be cancelled down to 2!.

## Factorial Interactive

Instructions: You can use the following interactive to find the factorial of any positive integer up to 30.

For numbers greater than 22!, you'll see output something like this: 2.652528e+32. The "e" stands for exponential (base 10 in this case), and the number has value 2.652528 xx 10^32.

## Binomial Theorem Formula

Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n:

(a+b)^n= a^n+na^(n-1)b +(n(n-1))/(2!)a^(n-2)b^2 +(n(n-1)(n-2))/(3!)a^(n-3)b^3 +...+b^n

This can be written more simply as:

(a + b)n = nC0an + nC1an − 1b + nC2an − 2b2 + nC3an − 3b3 + ... + nCnbn

We can use the {::}^nC_r button on our calculator to find these values.

This can also be written nCr.

## Binomial Theorem Interactive

The following interactive lets you expand your own binomial expressions. It shows all the expansions from n=0 up to the power you have chosen.

In the first line of each expansion, you'll see the numbers from Pascal's Triangle written within square brackets, [ ].

The second line of each expansion is the result after tidying up.

Instructions: You can use letters or numbers within the brackets. The maximum power you can use is 6.

Here are the expansions:

### Example 2

Using the binomial theorem, expand (x + 2)6.

### Example 3

Using the binomial theorem, expand (2x + 3)4

### Example 4

Using the binomial theorem, find the first four terms of the expansion (2a-1/x)^11

## Binomial Series

From the binomial formula, if we let a = 1 and b = x, we can also obtain the binomial series which is valid for any real number n if |x| < 1.

(1+x)^n=1+nx+(n(n-1))/(2!)x^2+(n(n-1)(n-2))/(3!)x^3+...

NOTE (1): This is an infinite series, where the binomial theorem deals with a finite expansion.

NOTE (2): We cannot use the {::}^nC_r button for the binomial series. The {::}^nC_r button can only be used with positive integers.

### Example 5

Using the binomial series, find the first four terms of the expansion sqrt(4+x^2).

Here is the graph of what we just did in Example 5. The darker colored curve is

y_1=sqrt(4+x^2)

The lighter colored one is the first 4 terms of the series we found, that is:

y_2=2+x^2/4-x^4/64+x^6/512.

The approximation is quite good between −2 < x < 2, but we would need to take many more terms for a good approximation beyond these bounds.

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