Exercise 1.

Find the resolvent cubic polynomial for the depressed quartic equation

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

Answer.

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

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

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Since

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adding the terms tex2html_wrap_inline109 on both sides yields

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The right side is a quadratic polynomial in x:

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

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The cubic polynomial

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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:

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The right side is a perfect square:

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

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This quadratic equation has the complex roots

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

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This quadratic equation has the real roots

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

Thus the quartic equation

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has the four roots:

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Helmut Knaust
Mon Jun 30 14:14:20 MDT 1997

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