It can be shown that the cubic polynomial equation
x3−px−q =0
will have real roots if the inequality p3/q2 > 27/4 is satisfied. Select five pairs of values for p and q for which this inequality is satisfied and hence, using the MATLAB function roots, verify in each case that the roots of the equation are real. In the 16th century the mathematician Ioannes Colla suggested the following problem: Divide 10 into three parts such that they shall be in continued proportion to each other and the product of the first two shall be 6. Taking x, y, and z as three parts, this problem can be stated as:
x + y + z =10, x/y=y/z, xy=6
Now by simple manipulation these equations can be expressed in terms of the specific variable y as:
y4+6y2−60y+36=0
Clearly if we can solve this equation for y then we can easily find the other variables x and z from the original equations. Use the MATLAB function roots to find values for y and hence solve Colla’s problem.
