Therefore the transformed equation is obtained from the original equation by multiplying the second term by the given factor, the third term by the square of this factor, and so on.
582. The chief use of this transformation is to clear an equation of fractional coefficients.
Ex. Remove fractional coefficients from the equation
Put x ={y}{q} and multiply each term by q^2 ; thus By putting q= 4 all the terms become integral, and on dividing by 2, we obtain ys ^Sy- -y + Q = 0.
583. To transform an equation into another whose roots exceed those of the original equation by a given quantity.
Let f(x) = 0 be the equation, and let h be the given quantity. Assume y = x + h, so that for any particular value of x, the value of y is greater by h ; thus x = y — h, and the required equation is f(y — h) = 0.
Similarly if the roots are to be less by h, we assume y =x — h, from which we obtain x = y + h, and the required equation is f(y +h) = 0.
584. If n is small, this method of transformation is effected with but little trouble. For equations of a higher degree the following method is to be preferred :
Let
put x = y + h, and suppose that f(x) then becomes
^02/" + ^i2/"~' 4- ^2^"' + • • • + gn-iV + g„. Now y = x — h; hence we have the identity
= ryo(x- - hy + q,(x - hf-' + • • • + fy„_ ,{x - h) + q„ ;
therefore q_n, is the remainder found by dividing f(x) by x - h; also the quotient arising from the division is
q,(x - hy ' + qi(:x - hy - + ••. + ^,._i.
