quotient. Raising, now, the oblique columns, we have the following concise form for the Here, we use only the second term of the divisor, with its sign changed; each term of the quotient is the sum of the vertical column under which it stands, and each term of the second line is obtained by multiplying the next preceding term of the quotient by the divisor, as written. Synthetic division is especially convenient when there are several successive operations in which each quotient becomes in turn a dividend. 465. The transformation of equations into others whose roots shall differ from those of the given equation by a given quantity, may now be effected by the process of synthetic division, or otherwise. EXAMPLES. 1. Transform the equation x3+x2-10x+8=0 into another whose roots shall be less by 4. 1st quotient, 1 + 5 +10, +48, 1st remainder, or v. 2d quotient, +4+36 19, +46, 2d remainder, or u'. 3d quotient, 1, 13, 3d remainder, or t'. Hence, y3+13y2+46 y+48=0 is the required equation. 2. Transform the equation x1 2 x - 10 into Therefore, 8y+23y2 30y+150 is the required equation. In the foregoing solution the coefficient 1 is not repeated, and the columns of figures are separated; in other respects it is like the preceding one. This method may also be applied when the coefficient of the first term of the given equation is not unity. 3. Increase the roots of the equation 6x3 11 x2 + 6 x -10 by 0.5. Therefore, 6-20 y2+21.5 y7.50 is the required equation. 4. Transform the equation - 9 x2 + 26 x 24 = 0 into another whose roots shall be less by 1. Ans. y36y2 + 11 y — 6 = 0. 5. Diminish the roots of the equation - 13x2+56 x Ans. y4y2+5y—2—0. - 800 by 3. 6. Increase the roots of x 5 x2+4=0 by 2. 466. To transform a complete equation into another whose second term shall be wanting. To make p'=0, it is only necessary to make nr+p=( p (Art. 462), or r= for x; that is, We must, then, substitute y n Р n Substitute for the unknown quantity a binomial consisting of a new unknown quantity, and the coefficient of the second term, taken with a contrary sign, and divided by the number denoting the degree of the equation. This substitution may be effected in either of the ways mentioned in Arts. 462-465. It is evident that the sum of the negative roots of the transformed equation must be equal to the sum of the positive roots (Art. 442). EXAMPLES. Transform the following equations into others whose second terms shall be wanting. Hence, the transformed equation is y3-3 y — 4 = 0, whose roots are less by 2. 2. —pz + q = 0. 3. Ans. — p2+q=0. +62-3x+4=0. Ans. y-15y+26= 0. 4.-8x+19—12x=0. Ans. -5y+4=0. 5. 8x + 23x2 — 30 x + 15 = 0. 6. x2+x2+4=0. Ans. -y-2y-10. Ans. yy += 0. DERIVED POLYNOMIALS. 467. A DERIVED POLYNOMIAL is one that may be obtained from any given polynomial, containing only a single unknown quantity, as x, by multiplying each term by the exponent of x in that term, and then diminishing the exponent by 1. The second derived polynomial is the derived polynomial of the first derived polynomial; the third is derived in like manner from the second, and so on, each being one degree lower than the preceding. Derived polynomials are also called derivatives. 468. A DERIVED EQUATION is one whose first member is a derivative of the first member of another equation. The first derived equation is also called the limiting or separating equation, when used to separate, or determine the limits of the roots of an equation. EXAMPLES. 1. Obtain the successive derived polynomials of x3+5x2 +3x+9. As 9 may be regarded as 9x, we have 0 × 9 for a |