POSITIVE INTEGRAL EXPONENT.
398. The Binomial Theorem, discovered by Kewton, is a
formula, by means of which any binomial may be raised to any
required power, without going through the process of invo-
lution.
399. To prove the Theorera for a positive integral ex'
portent.
By actual multiplication we may show that

(a -f a;)2 = a« + 2 a « + a?*

(a -I- a;)» = a« -f 3 a« x + 3 a X* + «■

(a + ar)* = a* + 4 a«ar -f 6 a"a;^ + 4 a a" -h SB*

In these results we observe the following laws :

1. The number of terms is one more than the exponent of the
binomiuL

2. The exponent of a in the first term is the same as the
exponent of the binomial, and decreases by one in each sue-
eeedinrj term,,

3. The exponent ofx in tJie second term is unittf,and m-
creases by one in each succeeding term.
BINOMIAL THEOREM. 299

4 Tlie coefficient of the first term is unity; and of the see-
mid terrriy is the exponent of the binomial,

6. If the coefficient of any term be multiplied by the expo-
nent of a in that term, and the product divided by the number
of the term, beginning at the left, the result will be the co-
efficient of the next term^

Assuming that the laws hold for any positive integral expo-
nent, n, we have
/ X 1 nCn—l) „ „ n(n—l)(n—2) . ,

This result is called the Binomial Theorem.
