Binomial theorem

The  exercise appears under the Algebra II Math Mission. This exercise introduces the binomial theorem as a method of expanding binomials raised to integer powers.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find number in square and use to find coefficient: This problem has a copy of Pascal's triangle and a missing space. The student is asked to find the missing number and used it to find a particular coefficient in a binomial expansion.Binthm1.png
 * 2) Use Pascal's triangle to find coefficient: This problem has a copy of Pascal's triangle and question about a binomial expansion. The student is asked to use the triangle to find a particular coefficient in a binomial expansion.Binthm2.png

Strategies
Knowledge of combinations or Pascal's triangle are encouraged to ensure success on this exercise, but a working knowledge of the Binomial Theorem or multiplication and patience are all that is necessary.
 * 1) In Pascal's triangle, each square is the sum of the two squares immediately above it.
 * 2) In general, the mth term (counting from zero) of (ax+by)^n is nCm (ax)^(n-m)(by)^m.

Real-life Applications

 * 1) The binomial theorem has many applications in combinatorics as a counting strategy.
 * 2) The binomial coefficients are related to Pascal's triangle which has many useful properties in number theory and applications in fractal geometry.