# Binomial Theorem

### terms equation term expansion

The binomial **theorem** provides a simple method for determining the coefficients of each term in the expansion of a binomial with the general equation (A + B)^{n}. Developed by Isaac Newton, this theorem has been used extensively in the areas of probability and **statistics**. The main argument in this theorem is the use of the combination formula to calculate the desired coefficients.

The question of expanding an equation with two unknown variables called a binomial was posed early in the history of **mathematics**. One solution, known as **Pascal's triangle**, was determined in China as early as the thirteenth century by the mathematician Yang Hui. His solution was independently discovered in **Europe** 300 years later by Blaise Pascal whose name has been permanently associated with it since. The binomial theorem, a simpler and more efficient solution to the problem, was first suggested by Isaac Newton. He developed the theorem as an undergraduate at Cambridge and first published it in a letter written for Gottfried Leibniz, a German mathematician.

Expanding an equation like (A + B)^{n} just means multiplying it out. By using standard **algebra** the equation (A + B)^{2} can be expanded into the form A^{2} + 2AB + B^{2}. Similarly, (A + B)^{4}can be written A^{4} + 4A^{3}B + 6A^{2 }B^{2} + 4AB^{3 }+ B^{4}. Notice that the terms for A and B follow the general pattern A^{n}B^{0},A^{n-1}B^{1},A^{n-2}B^{2},A^{n-3}B^{3},...,A^{1}B^{n-1}, A^{0}B^{n}. Also observe that as the value of n increases, the number of terms increases. This makes finding the coefficients for individual terms in an equation with a large n value tedious. For instance, it would be cumbersome to find the **coefficient** for the term A^{4}B^{3} in the expansion of (A + B)^{7} if we used this algebraic approach. The inconvenience of this method led to the development of other solutions for the problem of expanding a binomial.

One solution, known as Pascal's triangle, uses an array of numbers (shown below) to determine the coefficients of each term.

This triangle of numbers is created by following a simple rule of **addition**. Numbers in one row are equal to the sum of two numbers in the row directly above it. In the fifth row the second term, 4 is equal to the sum of the two numbers above it, namely 3 + 1. Each row represents the terms for the expansion of the binomial on the left. For example, the terms for (A+B)^{3} are A^{3} + 3A^{2}B + 3AB^{2 }+ B^{3}. Obviously, the coefficient for the terms A^{3} and B ^{3 }is 1. Pascal's triangle works more efficiently than the algebraic approach, however, it also becomes tedious to create this triangle for binomials with a large n value.

The binomial theorem provides an easier and more efficient method for expanding binomials which have large n values. Using this theorem the coefficients for each term are found with the combination formula. The combination formula is

The notation n! is read "n factorial" and means multiplying n by every positive whole integer which is smaller than it. So, 4! would be equal to 4 × 3 × 2 × 1 = 24. Applying the combination formula to a binomial expansion (A + B)^{n}, n represents the power to which the formula is expanded, and r represents the power of B in each term. For example, for the term A^{4}B^{3} in the expansion of (A + B)^{7}, n is equal to 7 and r is equal to 3. By substituting these values into the combination formula we get 7! / (3! × 4!) = 35, which is the coefficient for this term. The complete binomial theorem can be stated as the following:

See also Factorial.

## Resources

### Books

Dunham, William. *Journey Through Genius.* New York: John Wiley & Sons, 1990.

Eves, Howard Whitley. *Foundations and Fundamental Concepts of Mathematics.* NewYork: Dover, 1997.

Larson, Ron. *Precalculus.* 5th ed. New York: Houghton Mifflin College, 2000.

Perry Romanowski

## User Comments

about 9 years ago

Thanks so much for posting this explanation. Very clear and to the point. I understood it right away.

over 9 years ago

This is so simple and great! Thanks! I understood it right away...almost. ;)

over 10 years ago

I fell asleep during the class that was teaching this and the test is tomorrow. I'll now do fine on this section