Question

Based on Fibonacci's sequence, prove that the equation x^3 + 2x^2 + 10x = 20 can have no solution in the form a + rad(b), where a and b are positive rationals.

Answer 1

Fibonacci was his solution of the cubic equation

x^3+ 2x^2+ 10x = 20 which was reportedly given to him as a challenge.The accuracy of the result, it is worth

noting how, despite his writings on the Hindu – Arabic numeration system, he (and others, including Arabic mathematicians) still wrote fractional values in the Babylonian sexagesimal notation). In his analysis of this equation he also made an important observation which foreshadowed the nineteenth century results on the impossibility of

trisecting angles and duplicating cubes by means of straightedge and compass. It appears that the original version of the problem was to find a root of the given cubic

equation by means of classical Greek straightedge and compass methods. Fibonacci proved that the root could not be obtained by such methods.Rather than attempt to give Fibonacci’s proof, we shall analyze the equation from the

same viewpoint we employed to study the construction problems. As in those cases, the proof that a root cannot be found using straight edge and compass depends upon

showing that the polynomial x^3+ 2x^2+ 10x – 20 cannot be factored into a product of two rational polynomials of lower degree, or equivalently it does not have an integral factorization of this sort. If it had such a factorization then it would have a linear factor and hence an integral root. Furthermore, if it had an integral root then this root would have to divide 20 and thus would have to be one of the following:

±1, ±2, ±4, ±5, ±10, ±20

One is then left to check that none of these twelve integers is a root of the given polynomial. Direct substitution is perhaps the most immediate way of attacking this problem, but one can also dispose of many possibilities simultaneously by noting that the polynomial under consideration is positive for all integers x > 1 and it is negative for all integers x < 0.