ft * . 
BINOMIAL. 
general theorem for raising a quantity con- 
sisting of two terms to any power »n. 
The same general theorem will also serve 
for the evolution of binomials, because to 
extract any root of a given quantity 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 extract- 
ed. Thus, to extract the square root of 
« -(- ft, is to raise a -)- 6 to a power whose 
exponent is t. Novi', a 6 being found as 
above ; supposing m= a, you will find a-j-fti 
= aj-|-ix a-H-ftx — J X a~i b 2 - f. 
2 X 4 X 2 ® 2 ”i“> &C. ~ 82 -f- 
tities a, ft, c, Sec. B is the sum of the pro- 
ducts of every two ; C is the sum of the pro- 
ducts of every three, &c. &c. 
Let a =:6 = cr=:d = &c. then A, or a -|- 6 
+ e + d + &c -==»«; = aft-f-acB -f-ftc 
+ &c. =za 2 X the number of combinations 
of a, 6 , c, d, See. taken two and two, = n . 
n — 1 „ . 
; in the same manner it appears 
2 
that C = n . 
- 1 n — 2 
a 3 , Sec. And 
x-f-a.x-j-ft.x-J-c. &c. to n factors : 
; therefore x -f- a I’ = x” -|~ nax” - 1 
1 — 1 2 . n — 1 n — 2 . 
■ arx n - 2 -\-n . 
V 
8 16 af ' 
ft 3 
, & c. 
To investigate this theorem, suppose n 
quantities, x -fa, x -f ft, x c, &c. multi- 
plied together ; it is manifest that the first 
term of the product will be X 1 , and that 
x n — x n - 2 , &c. the other powers of x wall 
all be found in the remaining terms, with 
different combinations of a, 6 , c, d. &c. 
2 ~ ~ 1 " • 2 ‘ 3 
a;'!- 3 -|-&c. 
This proof applies only to those cases in 
which a is a whole positive number; but 
the rule extends to those cases in which n is 
negative or fractional. 
JEx.l. a-|-xl 8 =:a 3 -{-8a’x-|-28a 6 x 2 -f- 
56 a 4 x 3 -f- 70 a 4 x 4 -j- 56 a 3 x 5 + 28 a 2 x 6 -j- 
8 ax 7 -j-x a . 
Let x-f-ft.x-J-c.x-f-d. Sec. = x " - 
Px”- 
' + 
~ 2 + Q x”- 3 -(-&c. andx + a.x + 6 . -\-n. 
Ex. 2. l-f-xl” = l+KX-f M 
1 n — 2 
n — 1 
x-j-c. x-j-d.&c. = x"-f- Ax”- 1 -)- Bx” — 2 
then x”-f- Ax” — 1 Bx ”— 2 -f-&c. 
4-Qx”- 3 ^: 
2 
r X s -J- &c. 
and x-\-a x x n ~ l Ex 
Sec. or, 
x” 4- P t” - 1 4- Q x” - 2 4- &c. ? are the same 
ax”- 1 4-aPx”- 2 -f-&c. ^ series; 
therefore, A = P -)- a, B = Q -f- a P, &c. 
that is by introducing one factor, x-\-a, into 
the product, the coefficient of the second 
term is increased by a, and by introducing 
x -f 6 into the product, that coefficient is 
increased by 6 , Sec. therefore the whole 
value of A is a -f- ft + c + d -f &c. Again, 
by the introduction of one factor, x -f- a, the 
coefficient of the third term, Q, is increased 
by a P, i. e. by a multiplied by the preceding 
value of A, or by a x ft-j-c-f d^ Sec. and 
the same may be said with respect to the 
introduction of every other factor; there- 
fore upon the whole, 
Ex. 5. a 2 -|- x 2 1” — a 2n -\-na‘ n ~ 
a 2 ”— 4 x 4 -J- &c. 
2 -{-«. 
If either term of the binomial be negative, 
its odd powers will be negative, and conse- 
quently the signs of the terms, in which 
those odd 
changed. 
powers are found, will be 
Ex. 4. a — x ! 3 = a 8 — 8 a 2 x -f- 28 a 6 x 2 
— 56 a 3 x 3 70 a 4 x 4 — ■ 56 a 3 x 3 -j- 28 a 2 x‘ 
— 8 ax 2 -j-x 
a 2 — x*l ” =r a 2 ” n a 2 * -* 2 x 2 4 . 
« 2 ”— 4 x 4 — Sec. 
B — a . ft -j- v d -j- Sec. 
ft . c-f- d -f Sec. 
+ c . d -|- &c. 
In the same manner, 
If the index of the power to which a bino- 
mial is to be raised, be a whole positive num- 
ber, the series will terminate, because the 
1 — 1 n — 2 
2 ’ 3 
coefficient n 
. &c. will be- 
C =£ a . 6 . c . -f- d -j- &c. 
-J- a . c . cl -|- &e. 
-f- ft . c . u-j- &c. 
and so on ; that fa, A is the sum of the quaa- 
come nothing when it is continued to a -j -1 
factors. In all other cases the number Of 
terms will be indefinite. 
When the index is a whole positive num- 
ber, the coefficients of the terms taken 
ShIHhHBI 
:t~ icA 
WBHbSR. 
wa W 
m 
