607. Let us determine the number and situation of the real roots of a8 + 38a°-—9x#—4=0.
Here f_1(x)= 32° + 6% —9.
Now any positive factor may be introduced or removed in finding f_2(x), f_3(x), etc., for the sign of the result is not affected by so doing; hence multiplying the original equation by 3, we have
3a? + 62 —9)30° 4+ 9a? — 27x — 12(¢41 8e+62— 92
38a? — 18% —12
32+ 6Ga— 9
3) —24e— 3 = BrSl 2 OS ert il.
8x 4+1)24a? + 48e— T2324 5 24ar+ Sa 9)452— 12
5a— 8 ace 40x — 64 40x4+ 5
69. f_3(x) = 69.
We therefore have
f@)= 8 +32? — 9x —4,
fiz) = 32° + 6x—9,
f(x) =82+1,
i3@)— 695
Ae) Filet) AA) Fil) +
When x=-\infty we have —- + — 3 variations. When x=+\infty we have + + + + no variations. When x=0 we have — — + +4 _ 1 variation.
Hence the number of real roots is 3, of which one is positive and two are negative.
To determine the situation of these roots we substitute different numbers, commencing at 0 and working in each direction, thus:
