Unknown

1.4. Equations of Low Degree^21

(b) Show that the quartic polynomial in (a) can be written as the

product of two factors

(t” + ut + v)(t2 - ut + w)

where u, 21, w satisfy the simultaneous system

v+w-?.62=p

u(w - v) = q

vw=r.

Eliminate v and w to obtain a cubic equation in u2.

(c) Show how any solution u obtained in (b) can be used to find all

the roots of the quartic equation.

(d) Use Descartes’ Method to solve

t4 + t2 + 4t - 3 = 0

t4 - 2t2 + 8t - 3 = 0.

12. The qua&c equation: Ferruri’s method.
    (a) Let a quartic equation be presented in the form

t4 + 2pP + qt2 + 2rt + s = 0.

The strategy is to complete the square on the left side in such

a way as to incorporate the cubic term. Show that the equation

can be rewritten in the form

(t2 + pt -t u)2 = (p” - q + 2u)P + 2(pu - r)t + (I62 - s),

where u is indeterminate.

(b) Show that the right side of the transformed equation in (a) is the

square of a linear polynomial if u satisfies a certain cubic equa-

tion. Explain how such a value of u can be used to completely

solve the quartic.

(c) Use Ferrari’s Method to solve

t4 + t2 + 4t - 3 = 0

t4 - 2t3 - 5t2 + lot - 3 = 0.

13. Reciprocal equations. A reciprocal polynomial has the form

uxn + bx”-’ + CX”-~ + .+. + cz2 + bx + a,

in which Q # 0 and the coefficients are symmetric about the middle

one. A reciprocal equation is of the formp(t) = 0 with p(t) a reciprocal

polynomial.