Question

1. If p(x) is a complex polynomial with real coefficients, it is well known that it can be factored into a product of linear and quadratic terms with real coefficients, or into a product of linear terms only if the coefficients are allowed to be complex.

   First, use Maple to write q(z) = x5 −3x4 −3x3 +9x2 −10x+30 as a product of exact linear and quadratic factors with real coefficients. By exact, I mean you should leave any non-rational factors expressed as radicals; do not approximate terms like√3 as 1.73205, etc.

Then write q(x) as a product of only linear factors (which will involve complex numbers). Finally, do the same for product p(x) = x5 − 2x4 − 10x3 + 20x2 − 16x + 32.
Hint: Note that this question asks for four different answers, two for each polynomial. While the maple command factor is relevant, it will need a little assistance to be able to answer all four parts. See also RootOf and maybe convert,radical. Alternatively, there are other ways to do this. For example, using product or PolynomialTools.

Use maple to do this question,

Answer 1