#### Exercise 1.

Find the resolvent cubic polynomial for the depressed quartic equation

Check that z=3 is a root of the resolvent cubic for the equation, then find all roots of the quartic equation.

First, we move the x-term to the right side:

Luckily, the left side is already a perfect square, so our equation looks like

Since

adding the terms on both sides yields

The right side is a quadratic polynomial in x:

It will become a perfect square when its discriminant is 0. This means we want to find z so that

The cubic polynomial

is the cubic resolvent; it is easy to check that z=3 is indeed a root of the cubic resolvent. Using the value z=3 in Equation (1), we obtain:

The right side is a perfect square:

Next, we take square roots, discarding one of the choices:

This quadratic equation has the complex roots

If we regard, on the other hand, the second choice of sign, we obtain the quadratic equation

This quadratic equation has the real roots

(Alternatively, 2 more roots can be found by polynomial long division.)

Thus the quartic equation

has the four roots:

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