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.

LiveMath Solution

Computers can do this for us very easily. Let's get LiveMath to expand the binomial for us.

LIVEMath

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 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'

The LiveMath expansion of factorials is usually better than our calculators, in that it can go higher. You can go up to about 170!.

LIVEMath

Examples:

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

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

math expression

Note : cannot be cancelled down to 2!

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:

This can be written more simply as:

math expression

We can use the button on our calculator to find these values.

LiveMath can also find just the coefficients [numbers at the front] for us, too.

These are usually written nCr.

This LiveMath example will find the coefficients nCr. Your calculator can also do the same thing.

LIVEMath

Example 1:

Using the binomial theorem, expand .

Answer

Example 2:

Using the binomial theorem, expand

Answer

Example 3:

Using the binomial theorem, find the first four terms of the expansion

Answer

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.

math expression

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

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

Example:

Using the binomial series, find the first four terms of the expansion .

Answer

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3. Infinite Geometric Series

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Series and the Binomial Theorem

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4. The Binomial Theorem

LiveMath Solution

Pascal's Triangle

The Binomial Theorem

Properties of the Binomial Expansion (a + b)ⁿ