This equation is clearly satisfied when k^2-4=0, or k=\pm 2. It will be sufficient to consider one of the values of k; putting k = 2, we have
m+l=2, m-l=4; that is, l=-l, m=3.
Thus x^4 -2x^2 +8x-3=(x^2+ 2x -1)(x^2-2x+3);
hence x^2+ 2x -1=0, and x^2-2x+3=0;
and therefore the roots are - 1 \pm 2, 1 \pm -2.
624. The general algebraic solution of equations of a degree higher than the fourth has not been obtained, and Abel’s demonstration of the impossibility of such a solution is generally accepted by mathematicians. If, however, the coefficients of an equation are numerical, the value of any real root may be found to any required degree of accuracy by the method of Art. 626.
EXAMPLES XLVIII. k.
Solve the following equations :
1. x^3 - 18x = 35. 2. x^3+ 72x -1720=0. 8. x^3 + 63x - 316 = 0. 4. x^3+ 21x + 342 =0. 5. 28x^3 -9x^2+1=0. 6. x^3 - 15x^2 - 33x + 847=0. 7. 2x^3+3x^2+ 3x+1=0. 8. x^3 -6x^2 + 38x -18=0. 9. 8x^3 -36x+27=0. 10. x^3 -15x-4=0. 11. x^4 +8x^3 + 9x^2-8x-10=0. 12. x^4 +2x^3 -7x^2-8x+12=0. 13. x^4-3x^2-6x-2=0. 14. x^4 -2x^3 - 12x^2+ 10x+3=0.
INCOMMENSURABLE ROOTS.
625. The incommensurable roots of an equation cannot be found exactly. If, however, a sufficient number of the initial figures of the root have been found to distinguish it from the other roots we may carry the approximation to the exact value to any required degree of accuracy by a method first published in 1819 by W. G. Horner.
