of the quotient by means of the first term of the divisor with its sign unchanged. To avoid this liability, recollecting that the first term in each successive dividend is always cancelled by the product of the first term of the divisor by the corresponding term of the quotient, we retain the first term of the divisor with its sign unchanged, and change all the rest. The operation will then stand thus: 1-510-105-111+2-1 The work may be written more concisely thus: 11-510-10+5-1 The divisor is placed at the left of the dividend in a vertical column. Beneath, in a horizontal line, are placed the first terms of the successive partial dividends; and under the whole is written the quotient also in a horizontal line. The partial products are written under the terms of the dividend to which they belong, in a diagonal line from the left downwards toward the right. 23* 0 10 0 0 0 0 0 -618-50+124-289-250+629. 1-39-25+ 62. The operation, it is evident, terminates when the partial products have reached the right hand column. This is the case, in the present example, when the term 62 of the quotient is obtained. And since the columns to the right of this do not, when added, severally reduce to 0, there will be a remainder, of which the sums of these columns respectively will be the coefficients. Supplying the letters, we shall have, therefore, a1-3 a3 +9 a2-25 a +62 for the quotient, with a remainder - 289 a2250 a +629. 3. Divide 26-5x+15x-24x3+27 x2-13x+5 by 24-2x3+4x2-2x+1. Ans. x-3x+5. 4. Divide x+2x1y + 3 x3y2 — x2y3 — 2 x y — 3 y3 by x2 +2xy +3y2. Ans. 23 — y3. The process with the modification above is called Synthetic Division. The examples, art 39, will furnish an additional exercise for the learner. GENERAL PROPERTIES OF EQUATIONS. 225. Any expression which involves a quantity is called a function of that quantity. Thus, x2 + px, a x2 + b, (a + x)3 are all functions of x. In like manner, a x2 - by2, x2y+yx, are functions of x and y. 2. A function is usually indicated by some one of the letters, f, F, &c., the quantity or quantities of which the expression is a function being inclosed in a parenthesis. Thus, f (x) indicates a function of x, f (x, y) a function of x and y. 3. If by f (x) we denote a particular function of x, then ƒ (a) will denote the same function of a. Thus, if the first function is x2+5x+6, the second will be a2+5a+6. 4. It will be recollected that by the root of an equation we understand any quantity which, being substituted in the equation, will satisfy its conditions. 5. An equation of the second degree is sometimes called a quadratic equation, one of the third degree a cubic, and one of the fourth a bi-quadratic equation. 6. A complete equation of the nth degree with one unknown quantity, n being an entire and positive number, may be reduced to the form, x2+ A x2-1 +- В x2¬2 + С x2¬3 + which the coefficients A, B, C . Tx+U=0, in T, U, are any num bers whatever, positive or negative, entire or fractional. Every equation of this description, since it is supposed to be derived from a problem with sufficient and properly limited conditions, may be assumed to have at least one root. We now proceed to investigate the general principles necessary to the solution of numerical equations of any degree. if a is a root of the equation, then the first member is divisible by x-a. For if the division is not exact, let Q be the quotient, and R the remainder arising from the division by x a; then we have 2+A 2-1 + . . . . Tx+U = Q(x − a) +R. (2) But the left hand member of this equation is equal to 0; and since a is by hypothesis a root of the equation, we have x = a, or x — a = 0, and the equation (2) reduces to 0=0+R, or R = 0, that is, there is no remainder, and the division is exact. 2. Conversely, if the first member of the equation (1) is divisible by x a, then a is a root of the equation. For Q being the quotient arising from the division by x-a, the equation Q(x-a) = 0, returns to which is satisfied by the value x = a; hence a is a root of the equation. In the solution of equations we have frequent occasion to ascertain, by trial, whether a particular number is a root of the equation. From the preceding principle it is obvious that this may easily be done by division. Ex. 1. To determine whether 4 is a root of the equation, 23-9x26x-24=0. Dividing by x-4, and performing the operation by synthetic division we have Ans. 4 is a root, and if the proposed be divided by x — - 4 the equation which results will be –5+6=0. Ex. 2. To determine whether 5 is a root of the same equation. Ans. 5 is not a root, since the division by x-5 leaves a remainder of 6. Ex. 3. Is 2 a root of the equation x3-7x+6=9? Ex. 4. Is 3 a root of the equation -6x+8x-16 NUMBER OF THE ROOTS. 227. In order to the solution of an equation, we must first determine the number of its roots. An equation of the second degree with one unknown quantity has, we have seen, two roots. We shall now show that every equation with one unknown quantity has as many roots as there are units in the highest power of the unknown quantity, and no more. Let a be a root of the equation since by the last article this equation is divisible by x-a, it A', &c., being the new coefficients which arise from the division. But this equation is satisfied by x-a=0, or by Let b be a root of this last equation, then we have (x —b) (1⁄2”—2 + A′′x”¬3 + which is satisfied by x - b=0, or by T′′x+U")=0, T"x+U'=0. Continuing the operation, it will be seen that for every new factor obtained, the exponent of x is made one less, and that we shall have finally " + Ax-1+B x2+ Tx+U (x-p), in which the b, &c., is equal to n or to the number of units in the index of the highest power of the unknown quantity. And since there are as many roots as factors, there will be as many roots as units in the highest power of x, the unknown quantity. An equation, moreover, cannot have a greater number of roots than there are units in the highest power of x. Let V2"+A+Bx2+ = the roots of which are a, b, с V = (x − a) (x b) (x — c). |