ber of factors must be even. In this case the given condition is satisfied if c, d, e,---k are all greater than a, or less than b; thus it does not necessarily follow that f(a)=0 has a root between a and b.
EXAMPLES XLVIII. f.
1. Find the successive derived functions of 2«*—«3—2224+5a—-1.
Solve the following equations which have equal roots :
2. of 902? +40e4+12=0. 8 at— 6034 12e2?-10%4+3=0. 4. 0 — 13a! + 6723 — 17142 + 216% — 108 = 0. 5. ® — 98 4 497-—3242=—0. 6. 8at +403 — 18224 lla —2=0.
4%. Show that the equation 1023 —17z2+«+6=0 has a root between 0 and — 1.
8. Show that the equation «* — 523+ 342 + 35% —70=0 has a root between 2 and 8, and one between — 2 and — 3.
. 9. Show that the equation x! — 1222+ 12% —8=0 has a root between — 3 and — 4, and another between 2 and 3.
10. Show that 2° + 5! — 2022 — 19% —2=0 has a root between 2 and 3, and a root between — 4 and — 5.
STURM’S THEOREM AND METHOD.
603. In 1829, Sturm, a Swiss mathematician, discovered a method of determining completely the number and situation of the real roots of an equation.
604. Let f(x) be an equation from which the equal roots have been removed, and let f(x) be the first derived function. Now divide f(x) by f_1(x), and denote the remainder with its signs changed by f_2(x). Divide f_1(x) by f_2(x) and continue the operation, which is that of finding the H.C.F. of f(x) and f_1(x), except that the signs in every remainder are changed before it is used as a divisor, until a remainder is obtained independent of x; the signs in this remainder must also be changed. No other changes of sign are allowed.
