Binomial Theorem (1130 AD)

Theorem: This theorem requires a bit of background. A binomial is any algebraic sum or difference of exactly two terms. The binomial theorem is a formula for simplifying positive-integer powers of a binomial, for example (2x + 3y²)¹⁵. The theorem uses so-called binomial coefficients. Most students first encounter these numbers in an investigation of Pascal's triangle. The first few rows of Pascal's triangle are presented here.

1
1   1
1   2   1
1   3   3   1
1   4   6   4   1

Notice that each internal number in the triangle is the sum of the two numbers directly above it to the left and the right. The triangle is extended by first writing down these sums, and then placing a 1 to the left of the sums, and another 1 to the right of the sums. Most calculators now have what is referred to as combinations key, which typically has the form _nC_k. This represents the number of combinations of n objects chosen k at a time. The last row given in the triangle above is actually a listing of ₄C₀, ₄C₁,up to ₄C₄. To see why ₄C₂ = 6, consider 4 objects, A, B, C, and D. ₄C₂ represents the number of combinations of these 4 objects taken 2 at a time. It is clear to see that AB, AC, AD, BC, BD, and CD represents an exhaustive list of all such combinations.

Given all of that, the binomial theorem is a formula for listing all (n+1) terms of the expansion of (a+b)ⁿ. The kth term, letting k range from 0 to n, is given by _nC_k X a^k X b^(n-k). So, in our example, if we wanted the ninth term in the expansion of (2x + 3y²)¹⁵ it would equal ₁₅C_n X (2x)⁹ X (3y²)⁶ = 5005 X 512x⁹ X 729y¹² = 1,868,106,240 x⁹ y¹².

Author: Chinese mathematicians are known to have used the binomial coefficients as early as 1100. Portions of the first proof of the general theorem date back to the middle 1500's. The proof's author is unknown.

Importance: Obviously, it allows for more rapid expansion of certain algebraic forms than would otherwise be possible, but applications of the binomial theorem are also frequently used in the proofs of other results. As one simple example, it is possible to apply the principle of induction as to the expansion of (1+1)n to establish that the sum of all binomial coefficients for a specific n must be a power of two. This can be supported by adding all of the numbers of any row of the triangle given above.