roots are known we may depress the equation to one of the (x-k)th degree. All the roots but two being known, the depressed equation is a quadratic from which the remaining roots are readily obtained.
569. Formation of Equations. Since f(x)=(x-a_1)(x-a_2) \ldots (x-a_n) [Art. 567], we see that an equation may be formed by subtracting each root from the unknown quantity and placing the continued product of the binomial factors thus formed equal to 0.
Ex. Form the equation whose roots are 1, -2, and {1}{2}.
EXAMPLES XLVIII. a.
1. Show that 4 is a root of x3 — 572-2x+4+24=0. 2. Show that + 3 is a root of 23+ 72?+72%+15=0. 3. Show that — } is a root of 623 + 1722-4x%—3=0. 4. Show that 4 is a root of 1023 -32? -97+4=0, 5.One root of #3 + 622 — 6x — 63 = 0 is 3; find the others. 6. One root of x? — 2322 + 1662 — 378 = 0 is7; find the others. 7. One root of 23 — 22? + 6x — 9,3, = 0 is 3; what are the others ? 8. Two roots of x! — 1572+ 10x”"+424=0 are 2 and 3; find the others. 9. Two roots of <t — 323 — 21724 43x + 60 = 0 are 3 and 5; find the others. 10. One root of x? + 2ar? + 5a" + 4a?=0 is — a; what are the others ? 11. Form the equation whose roots are — 1, — 2, and — 5. 12. Form the equation whose roots are — 2, — 3, + 3, and — }.
570. Relations between the Roots and the Coefficients. Let us denote the equation by
,
and the roots by a, b, c, \ldots k; then we have identically
