A Quintic Which is Solvable By Radicals

Let's look at x^5-1. Notice that this polynomial with a=1, f=-1, and all the rest of the quintic coefficients 0 has a real root at x=1.

> plot(x^5-1,x=-2..2);

Notice that this root is easily expressed in terms of a and/or f - for example

x1=a represents x1=1. We'd like to look at the other roots and see that they are also

expressable in terms of the coefficients a=1 and f=-1.

> simplify((x^5-1)/(x-1));

> g:=(x^2+((1+sqrt(5))/2 )*x +1)*(x^2+((1-sqrt(5))/2)*x+1);

> f:=unapply(expand(g),x);

I figured out how to write f as a product of two quadratic equations.

I did this via algebra: assume that

= f(x) = (x^2 +ax +1)(x^2+bx +1),

multiply everything out, and then solve for a and b.

You should check this for yourself!

Now I'm going to find the roots of the first factor of g using the quadratic formula (Maple will make this easy)

> solve(x^2+((1+sqrt(5))/2 )*x +1=0,x);

> one:=%[1];two:=%%[2];

Let's check to make sure these are roots:

> simplify(expand(f(one)));simplify(expand(f(two)));

Good, these are roots.

Now find a radical expression of the first root (one) in terms of a=1 and/or

f=-1:

Use Maple commands to find the roots of the second factor of g. Define these roots as three and four:

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Now list all 5 roots, which are all expressible in terms of the coefficients:

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