324 MATHEMATICS. 



multiplication ; and consists in cancelling from the dividend all the 

 factors which it has in common with the divisor; the remaining 

 factors being the quotient. Division of polynomials is performed in 

 much the same manner as arithmetical division ; requiring first that 

 all the terms both of the dividend and divisor should be arranged 

 according to the powers of some one letter ; after which the first term 

 of the quotient is fouad by dividing the first term of the dividend by 

 the first of the divisor. Of algebraic fractions, which are similar to 

 arithmetical, we have no room to speak farther. 



2. An Equation, is an expression denoting the equality of two 

 quantities : and a Simple Equation, is one in which no unknown 

 quantity is multiplied either by itself or by any other unknown 

 quantity. The sign of equality, (=), is read, equal to, and is placed 

 between the two equal qualities which are the first and second mem- 

 bers of the equation. Common algebraic problems are most fre- 

 quently solved by means of equations ; or by proportions, from which 

 equations are easily obtained. To form the equation, we usually 

 express the unknown quantity, if there be but one, by the letter x ; 

 and with this we form an expression which, by the conditions, is 

 equal to some other expression or formula ; after which it only 

 remains to find the value of x from the equation thus formed. Thus, 

 to find a certain number, twice which, being added to 76, and the sum 

 divided by 4, the quotient will be equal to 10 times the same number, 



we write the equation = IQ x ; as the first operation. 



If we multiply each member of the above equation by 4, it will 

 form another equation, free from denominators, and without changing 

 the value of x ; viz. 2 x -f 76 = 40 x. The next step, is, to bring 

 all the terms containing the unknown quantity to stand by themselves, 

 in one member, usually the first member of the equation. In the 

 present example, to transpose the term 2 x, to the second member 

 of the equation, we cancel it, where it stands, which is really sub- 

 tracting it from the first member : and hence we must also subtract it 

 from the second member; and write 76=40 # 2x; or by reduc- 

 tion, 76=38 x. If, now, we divide both sides of the equation by 

 38, the co-efficient of the unknown quantity, we shall have 2 = x ; 

 or x = 2. When the problem involves two distinct unknown quan- 

 tities, say x and y, there must be two distinct equations ; from one 

 of which we find the value of x, in terms containing?/; and then 

 substitute this value of x, wherever x occurs, in the other equation : 

 which will then contain only one unknown quantity, y. 



3. Quadratic Equations, are those which contain the square or 

 second power of the unknown quantity ; but no higher power. To 

 resolve them, we first transpose, if necessary, so as to bring all the 

 terms containing x* to stand first in order ; those containing x to 

 stand next ; and all the known terms, that is, those which do not 

 contain x, form the other member of the equation. We then divide 

 both members by the co-efficient of x z , which reduces the equation 

 to the regular form, x* = a, for pure quadratics, and x* -f ax = b, 

 for those which are affected, or complex: a and b here simply 

 denoting any known quantities. A pure quadratic, is then resolved. 



