Cubic Equation Cardano's Method Cube Roots Lagrange Resolvents

where we will assume that the coefficients a₀,...,a₃ belong to a field of characteristic other than two or three, with a₀ being non-zero.

Solving a cubic equation amounts to finding the roots of a cubic function. Every cubic equation with real coefficients has at least one solution x among the real numbers. We can evaluate the different possible cases in terms of the quantities

1 Cardano's method

The substitution x = t - a/3 eliminates the quadratic term and we get at a cubic equation of the form

The above system for u and v can always be solved: solve the second equation for v, substitute into the first equation, solve the resulting quadratic equation for u³, then take the cube root to find u. When the cubic equation has three real roots, the quadratic equation will give complex solutions, and so finding the real roots of a polynomial with real coefficients by means of extracting cube roots sometimes requires the use of complex numbers. This was already noticed by Cardano and is a strong argument for the usefulness (if not the existence) of complex numbers; historically the acceptance of complex numbers as having at least an imaginary sort of of existence on account of their usefulness stemmed from this fact.

Once the values for t are known, the substitution x = t - a/3 can be undone to find the values of x solving the original equation.

1.1 Cube roots

In order to consistently define cube roots, we take the principal branch of the cube root function; so that if we express a number in polar coordinates with the polar angle θ in the range , to define the complex cube root we take the cube root of the radius and divide the polar angle by three. This means that the cube root of a negative real number will be a complex number.

Note that in finding u, there were six possibilities, since there are two solutions to the square root, and three complex solutions to the cubic root. However, which solution to the square root is chosen does not affect the final resulting x.

2 Lagrange resolvents

The symmetric groupIn mathematics, the symmetric group on a set X denoted by S or Sym X , is the group whose underlying set is the set of all bijective functions from X to X in which the group operation is that of composition of functions, i. two such functions f and g can S₃ of order three has the cyclic groupIn mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that every element of the group is a power of a''. Equivalently, an element a of a g of order three as a normal subgroupIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t, which suggests making use of the discrete Fourier transformIn mathematics, the discrete Fourier transform (DFT sometimes called the finite Fourier transform is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. It is sometimes deno of the roots, an idea due to

LagrangeJoseph Louis Lagrange ( January 25, 1736 April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. Lagrange worked for Frederick II, in Berlin, for twenty years. It was Lagrange who developed the mean value theorem.

If r₀, r₁ and r₂ are the roots of equation (1), and if

so that ζ³=1 and ζ is a primitive third root of unityIn mathematics, the n th roots of unity or de Moivre numbers named after Abraham de Moivre (1667 1754), are complex numbers located on the unit circle. They form the vertices of a n sided regular polygon with one vertex on 1. Definition For a given n the, then we may set

The roots may then be recovered from the three s_i by inverting the above linear transformation, giving

We already know the value s₀ = -a, so we only need to seek values for the other two. However, if we take the cubes, a cyclic permutation leaves the cubes invariant, and a transpostion of two roots exchanges s₁³ and s₂³, hence the polynomial

is invariant under permutations of the roots, and so has coefficients expressible in terms of (1). Using calculations involving symmetric functions or alternatively field extensions, we can calculate (3) to be

then the roots of (4) are

and

Taking cube roots gives us u and v, which we can set to s₂ and s₃ and thereby obtain our solution.

3 Factorization

If r is any root of (1), then we may factor using r to obtain

Hence if we know one root we can find the other two by solving a quadratic equation, giving

for the other two roots. If we are finding the roots of a polynomial with real coefficients and one real root, we can find the real root purely in terms of the real (rather than complex) cube root function, or alternatively stated we can find the root by extracting cube roots only of positive quantities. The complex conjugate roots can then be found as above.

4 Chebyshev radicals

The cube root function is in some respects not a well-behaved function, or one convenient for the purposes of finding the roots of a cubic equation. While cube roots are well-known and traditional, it is possible to use other algebraic functions to determine the roots, and avoid some of the problems of cube roots. The cube root function has a branch singularity at zero, as a result of which the real cube root function does not extend nicely to a complex cube root function. Moreover, when using cube roots to find the roots of a polynomial with three real roots we must take the roots of complex numbers, which introduces complex numbers into a situation which does not, in fact, require them.

We can get around these problems by using Chebyshev cube roots in place of ordinary cube roots. The polynomial is the third Chebyshev polynomial normalized to the interval [-2, 2], which we use in place of [-1, 1] so as to obtain a monic polynomial. This polynomial function is equal to

which therefore can be inverted to obtain

which satisfies .

This procedure is precisely analogous to the definition of the cube root in terms of logarithms and exponentials, with 2 arccosh(x) in the place of ln(x), and cosh(x/2) in the place of exp(x). However, it tangles us up in the question of the branch cuts of arccosh, and rather than dealing with the complexities of that, we can define the Chebyshev cube root for all complex values directly. For , we have the following convergent series:

We can now define for the whole complex plane by taking a branch cut from to -2, and analytically continuing; continuing to the branch cut itself through the upper half-plane.

Now if we want the roots of we may find them in terms of as

If we have a cubic equation in depressed form, we may write it as . Substituting we obtain . From this we obtain solutions to our original equation in terms of the Chebyshev cube root as

. If now we start from equation (1) and reduce to the depressed form, we have

and , leading to

This gives us solutions to (1) as

Suppose the coefficients of (1) are real. If s is the quantity q/r from the section on real roots, then s = t²; hence 0 < s < 4 is equivalent to -2 < t < 2, and in this case we have a polynomial with three distinct real roots, expressed in terms of a real function of a real variable, quite unlike the situation when using cube roots. If s > 4 then either t > 2 and is the sole real root, or t < -2 and is the role real root. If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginay number; in this case is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

which is a real function of a real variable with no singularities along the real axis. If a polynomial can be reduced to the form with real t, this is a convenient way to solve for its roots.

5 See also

mathematics polynomials

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