It will be found on trial that the values a=-, b = do not satisfy the third equation 2a2b = - 32; hence we are restricted to the values a = 3, b=-.
Thus the roots are , , -
574. Although we may not be able to find the roots of an equation, we can make use of the relations proved in Art. 570 to determine the values of symmetrical functions of the roots.
Ex. Find the sum of the squares and of the cubes of the roots of the equation x3 — px2 + qx —r=0.
Denote the roots by a, b, c; then a+b+c=p, be+ca+ab=q.
Now a2 4+ 2+ 2 =(a+ b+ 0)? —2(be + ca + ab)= yr —2Q.
Again, substitute a, b, c for x in the given equation and add; thus
@+b84+8—p(e@+F+)+¢e(arb+c)—38r=0; - @+ 084 3= p(p?—-2¢)—pat+3r=p'—spqt 3r.
EXAMPLES XLVIII. b.
Form the equation whose roots are:
1. 2, 3, +. 2. 0,0, 2, 2. -3, -3. 3. 2,2, —2, —2, 0, 5. 4. a+b, a—b, —a+b, —a—b.
Solve the equations :
5. x¢—16 x3 + 8622 — 176x 4 105 =0, two roots being 1 and 7. 6. 423+ 1622 — 9x — 36 =0, the sum of two of the roots being
7. 423 + 2022 — 28x46 =0, two of the roots being equal. 8. 3828 — 2622 + 52x — 24 = 0, the roots being in geometrical progression. 9. 2x8 — x2 — 22x — 24 =0, two of the roots being in the ratio of 3:4. 10. 242° + 4622+ 9x—9=0, one root being double another of the roots. 11. 8xt— 223 — 2722 +62+9=0, two of the roots being equal but opposite in sign.
- A function is said to be symmetrical with respect to its variables
when its value is unaltered by the interchange of any pair of them ; thus x + y+ z, be +ca+ ab, x^3 + y^3 + z^3 - xyz are symmetrical functions of the first, second, and third degrees respectively. [See Art. 319.]
