BINOMIAL, in algebra, a root consist ing of two members, connected by the sign + or —. Thus a + b and 8-3 are binomials; consisting of the sums and differences of these quantities.
The powers of any binomial are found by a continual multiplication of it by it self. For example the cube or third pow er of a+b, will be found by multiplica tion to be a3+3 a'b+Sab'+b3 ; and if the powers of a—b are required, they will be foundthe same as the preceding, only the terms in which the exponent of b is an odd number willbe found negative. Thus the cube of a—b will be found to be a3— where the second and fourth terms are negative, the exponent of b being an odd number in these terms. In general the terms of any power of a—b are positive and negative by turns.
It is tobe observedthatin the first term of any power of a+b, the quantity a has the exponent of the power required, that in the following terms the exponents of a decrease gradually by 'the same differen. .
ces, viz. unit, and that in the last terms it is never found. The powers of b are in the contrary order ; it is never found in the first term, but its exponent in the se cond term is unit ; in the third term its exponent is 2 ; and thus its exponent in creases, till in the last term it becomes equal to the exponent of the power re quired.
As the exponents of a thus decrease, and at the same time those of b increase, the sum of their exponents is always the same, and is equal to the exponent of the power required. Thus in the sixth power of a+b, viz. 6+15 a4 P+20 a3 63 +15 b4+ 6 a + b3, the exponents of a decrease in this order 6, 5, 4, 3, 2, 1, 0; and those of b increase in the contrary or der 0, 1, 2, 3, 4, 5, 6. And the sum of their exponents in any term is always 6.
In general, therefore, if a+b is to be raised to any power m, the terms without their coefficients will be am, am '1,', am-3b3, am_414, am-5M , &c. conti nued till the exponent of b become equal to in.
The coefficients of the respective terms 2n-1 Xwill be 1 ; m ; in— 1 • m —X 2 m-2 m-1 m-2 m-3 ; m X X 3 X m-1 —2 m-3 X X 3X X &c.
continued until you have one coefficient more than there are units in in.
It follows therefore by these rules, that 772-1 In am 1 b + m+ - + M-1 m-2 LLm m X N X X am M-1 m-2 m-3 /03 + in X - T- X X x am--464 +, &c. which is the binomial or general theorem for raising a quantity consisting of two terms to any power m.
The same general theorem will also serve for the evolution of binomials, be cause to extract any root of a given quan tity is the same thing as to raise that quantity to a power, whose exponent is a fraction that has its denominator equal to the number that expresses what kind of root is to be extracted. Thus, to extract the square root of a + b, is to raise a+b to a power whose exponent is Now a + Oa being found as above ; suppos ing ni = you will find = X b-Fi a b' + X x a — b3 ±, &c. = a t+ b , b3 -t- &c.
8a4. 16af To investigate this theorem, suppose n quantities, x + a, x + b, x c, &c. multiplied together; it is manifest that the first term of the product will be xn, and that xn—t, xn-2, &c. the other pow ers of x, will all be found in the remain ing terms, with different combinations of a, b, c, d, &c.
Lets b .x c.x-Fd.&c + P xn-2 + Q xn-3 &c. and x + a . x b x c x-Ed gLe =- xn+ + B + &c then xn + A .0_1...E B xn--2+ &c. and -FQxn-3+ &c. or, xn+Pxn—t+Qxn Stc.Z are the axn—'-'&c.—'4-&c. sameseries; therefore, A=P + a, B=Q + a P, &c. that is, by introducing one factor, x±a, into the product, the co-efficient of the se cond term is increased by a, and by intro ducing x±b into the product, that co.e ffi cleat is increased by b, &c. therefore the whole value of A is a + b d &c. Again, by the introduction of one factor, x+a,,the co-efficient of the third term, Q., is in eased by a P, e. by a multiplied by tire preceding value of A, or by aX b d &c. and the same may be said with respect to the introduction of every other factor; therefore, upon the whole, B = a . b+c-H-1-&c.