Let the given equation be
_p„a^" + PiX""-^ + V^""'^^ H V Pn-i ^ + Pn = ;
then if y = x — h, we obtain the new equation
Po(y + hy +2h(y + ^0'"' -^I'^iv + ^O""' + ••• +k = o,
which, when arranged in descending powers of y, becomes
If the term to be removed is the second, we put np0h+p1=0, so that h = -{p1} {np0} ; if the term to be removed is the third, we put
'l^^pji'-^{n - l)pji +p, = 0,
and so obtain a quadratic to find h; and, similarly, we may remove any other assigned term.
Sometimes it will be more convenient to proceed as in the following example.
Ex. Remove the second term from the equation px^ + qoiP + rx -{- s = 0. q
Let a, &, c be the roots, so that a+b +c = - {q}{p} Then if we increase each of the roots by {q}{3p}, in the transformed equation the sum of the roots will be equal to -{q}{p} + {q}{p} ; that is, the coefficient of the second term will be zero.
Hence the required transformation will be effected by substituting x — {q}{3p} for x in the given equation. op
As the general type of a cubic equation can be reduced to a more simple form by removing the second term, the student should carefully notice that the transformation is effected by substituting x minus the coefficient of the second term divided by the degree of the equation, for x in the given equation.
