We may therefore enunciate Descartes’ Rule as follows:
An equation f(x)=0 cannot have more positive roots than there are variations of sign in f(x), and cannot have more negative roots than there are variations of sign in f(-x).
Ex. Consider the equation #9 + 528 — #3 4+ 74+4+2=0.
Here there are two changes of sign, therefore there are at most two positive roots.
Again f(-x)=— «9 + 528 + « — 7% +2, and here there are three changes of sign, therefore the given equation has at most three negative roots, and therefore it must have at least four imaginary roots.
593. It is very evident that the following results are included in the preceding article.
(i.) If the coefficients are all positive, the equation has no positive root; thus the equation a°+2°+2x4+1=0 cannot have a positive root.
(ii.) If the coefficients of the even powers of x are all of one sign, and the coefficients of the odd powers are all of the contrary sign, the equation has no negative root; thus the equation
ot +o —2e+o%—32?+7T2—5=—0
cannot have a negative root.
EXAMPLES XLVIII. e.
Find the nature of the roots of the following equations :
1. xt +203 — 1342 - 14x 4 24=0.
2. at — 1023 + 8522 — 50x 4+ 24=0.
3. 8a!4 12074+52—4=-—0.
4. Show that the equation 227 — «4 + 43 ~ 5=0 has at least four imaginary roots.
5. What may be inferred respecting the roots of the equation g0 — 496 +o! 24 —-3=0?
6. Find the least possible number of imaginary roots of the equation 2 —254+ ¢44+2241=0.
