12. 54 x^-S0x'^-26x+ 10 = 0, the roots being in geometrical progression. 13. 32 x^ — 48 x"2 + 22 ic — 3 = 0, the roots being in arithmetical progression, 14. 6x4 - 29x3 + 40x2 - 7 a: - 12 = 0, the product of two of the roots being 2. 15. X* - 2 x^ - 21 x2 + 22 X + 40 = 0, the roots being in arithmetical progression. 16. 27 x4 - 195 x3 + 494 x^ _ 520 x + 192 = 0, the roots being in geometrical progression. 17. 18x3 + 81 x2 + 121 X + GO = 0, one root being half the sum of the other two. 18. Find the sum of the squares and of the cubes of the roots of x* + qx^ + rx + s = 0.
575. Fractional Roots. An equation whose coefficients are integers, that of the first term being unity, cannot have a rational fraction as a root.
If possible suppose the equation lias for a root a rational fraction in its lowest terms, represented by {a}{b}. Substituting this value for x and multiplying through by b^{n-1} we have
Transposing,
This result is impossible, since it makes a fraction in its
lowest terms equal to an integer. Hence a rational fraction
cannot be a root of the given equation.
576. Imaginary Roots. In an equation with real coefficients imaginary roots occur in pairs.
Suppose that f(x) = 0 is an equation with real coefficients, and suppose that it lias an imaginary root a+ib ; we shall show that a-ib is also a root.
