Algebraical Formulæ and Rules

for the Use of Candidates for Responsions

Meaning of $a^{x}$	$a . a . a . &c.$ (`x` factors).
hence $a^{1}$ =	`a`
Addition, when terms containing same powers of same letters are added:—
(α) two terms only:—
(1) with same sign	add coefficients, and repeat sign.
(2) with different signs	take difference of coefficients, with sign of greater.
(β) many terms	collect coefficients of + terms into one, and those of − terms into one, and proceed as before.
Subtraction, when a term is taken from another containing same powers of same letters:—
(α) when minuend is greatest, and signs are same	subtract, and repeat sign.
(β) otherwise	change sign of subtrahend, and proceed as in addition.
Brackets, to put on or take off:—
(α) when sign outside is “+”	keep signs within unchanged.
(β) is “−”	change signs within.
Multiplication:—
(α) as to indices, when two or more powers of same letter are multiplied	add indices.
e. g. $x^{a} . x^{b} . x^{c} . &c. =$	$x^{a + b + c + &c.}$ .
(β) as to signs, when two terms are multiplied:—
(1) signs like	sign of answer is “+”.
(2) signs unlike	is “−”.
Division:—
(α) as to indices, when a power of a letter is divided by another power of the same letter	subtract index of divisor from index of dividend.
e. g. $x^{a} \div x^{b} =$	$x^{a - b}$ .
hence $x^{0} =$	1.
$x^{- a} =$	$\frac{1}{x^{a}}$ .
$x^{- 1} =$	$\frac{1}{x}$ .
(β) as to signs, when one term is divided by another	same rules as in multiplication.
Involution:—
(α) a mononomial:—
(1) as to indices	multiply each index by index of required power.
e. g. ${(x^{a} . y^{b} . z^{c} . &c.)}^{n} =$	$x^{a n} . y^{b n} . z^{c n} . &c.$
(2) as to signs:—
(a) when index of required power is even	sign “+”.
(b) when it is odd	sign same as given quantity.
(β) a binomial:—
1. ${(a + b)}^{2} =$	$a^{2} + 2 a b + b^{2}$ .
2. ${(a - b)}^{2} =$	$a^{2} - 2 a b + b^{2}$ .
3. ${(a + b)}^{3} =$	$a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ .
4. ${(a - b)}^{3} =$	$a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$ .
5. ${(a + b)}^{n} =$	$a^{n} + n . a^{n - 1} b + &c.$ Index of `a` decreases while that of `b` increases. Coefficient of each term is formed from preceding by multiplying together coefficient and index of `a` and dividing by place of term. Signs all “+”.
6. ${(a - b)}^{n} =$	$a^{n} - n . a^{n - 1} b + &c.$ , as before. Signs alternately “+” and “−”.
(γ) a quantity of 3 or more terms	collect the terms into 2 brackets, and proceed as before.
Evolution of a mononomial:—
(α) as to indices	divide each index by index of required root.
e. g. $\sqrt[n]{x^{a} . y^{b} . z^{c} . &c.} =$	$x^{\frac{a}{n}} . y^{\frac{b}{n}} . z^{\frac{c}{n}} . &c.$ .
hence $\sqrt[n]{x} =$	$x^{\frac{1}{n}}$ .
(β) as to signs:—
(1) when index of required root is even:—
(a) sign “+”	sign of answer is “+” or “−”.
(b) sign “−”	root does not really exist.
(2) when it is odd	sign same as given quantity.
Resolution of binomials, &c., into factors:—
general rule	divide out all mononomial factors, placing them outside a bracket. For the factor within the bracket try the following formulæ.
(α) a binomial:—
1. $a^{2} + b^{2}$	has no factors.
2. $a^{2} - b^{2}$ =	$(a + b) . (a - b)$ .
3. $a^{3} + b^{3}$ =	$(a + b) . (a^{2} - a b + b^{2})$ .
4. $a^{3} - b^{3}$ =	$(a - b) . (a^{2} + a b + b^{2})$ .
5. $a^{n} + b^{n}$ , (where `n` is an odd prime) =	$(a + b) . (a^{n} - a^{n - 1} b + &c.)$ Index of `a` decreases while that of `b` increases. Signs alternately “+” and “−”.
6. $a^{n} - b^{n}$ , (where `n` is an odd prime) =	$(a - b) . (a^{n} + a^{n - 1} b + &c.)$ , as before. Signs all “+”.
(β) a trinomial:—
1. $a^{2} + 2 a b + b^{2}$ =	${(a + b)}^{2}$ .
2. $a^{2} - 2 a b + b^{2}$ =	${(a - b)}^{2}$ .
3. $A x^{2} + B x y + C y^{2}$ , where $B^{2} - 4 A C$ is a positive square (call it $K^{2}$ ) =	$A . (x + \frac{B + K}{2 A} . y) . (x + \frac{B - K}{2 A} . y)$ .
G. C. M. and L. C. M. of any number of mononomials:—
(α) G. C. M.	take each factor that occurs in all, with lowest index it bears.
(β) L. C. M.	take each factor that occurs, with highest index it bears.
G. C. M. of binomials, &c.:—
(α) two quantities:—
(1) general rule	arrange both in order of indices of some one letter, bracketing coefficients of any terms which contain the same power of it: then divide greater by less, and divisor by remainder, and so on till there is no remainder: the last divisor is the G. C. M.
(2) particular rules:—
(a) when a factor is observed in one of the quantities	divide it out.
(b) when in both	divide out, and multiply the answer by it
(c) when first term of divisor will not exactly divide that of dividend	find L. C. M. of their coefficients, and multiply dividend by such a number as will raise coefficient of first term to this L. C. M.: but first try whether this multiplier, or any factor of it, will divide divisor.
(β) three or more quantities	find G. C. M. of the first two; then G. C. M. of answer and third quantity, and so on.
L. C. M. of binomials, &c.:—
(α) two quantities	product divided by G. C. M.
(β) three or more quantities	find L. C. M. of first two; then L. C. M. of answer and third quantity, and so on.