Complex polynomial equations: the Cardano and Ferrari formulas

Summary

This lesson studies polynomial equations over the field of complex numbers. From the fundamental theorem of algebra, which is taken as given, it deduces by induction that every polynomial of degree $n$ is a product of $n$ linear factors and has exactly $n$ roots counted with multiplicity; it proves Vieta’s formulas for any degree and the formulas for the sums of the squares and of the cubes of the roots. It solves binomial, trinomial and reciprocal equations by reducing them to others of lower degree. Then it reduces the cubic equation to the depressed form, derives Cardano’s formula, classifies the roots by means of the discriminant, shows with Bombelli’s equation why the formula forced the acceptance of complex numbers even when all the roots are real, and solves that case with the triple-angle formula. Finally, it reduces the quartic equation to the depressed form, proves Ferrari’s method, which splits it into two quadratic equations by means of a root of its resolvent cubic, and explains why no similar formula exists from degree five on.

Learning objectives

Deduce from the fundamental theorem of algebra the factorization of every complex polynomial into linear factors, and relate the roots to the coefficients by means of Vieta’s formulas and the power sums.
Solve binomial, trinomial and reciprocal equations by reducing them to equations of lower degree.
Reduce the cubic equation to the depressed form, derive Cardano’s formula and classify the roots of a real cubic according to the sign of its discriminant.
Solve a cubic with three real roots by the trigonometric method, and explain with Bombelli’s example the historical role of Cardano’s formula in the acceptance of complex numbers.
Solve quartic equations by Ferrari’s method, reducing them to the depressed form and factoring them by means of a root of the resolvent cubic.

The fundamental theorem of algebra over the complex numbers

In the lesson of the course Algebra and Geometry I on the roots of higher-degree polynomials, the fundamental theorem of algebra was stated, its consequence that every complex polynomial of degree $n$ is a product of $n$ linear factors was sketched, and from it was deduced that every real polynomial is a product of linear factors and irreducible quadratic factors; here the complete proof, by induction, of that factorization over the complex numbers is given. This lesson adopts the point of view of the complex numbers: the coefficients and the roots may be complex, and the question ceases to be whether an equation has a solution and becomes how to find all its solutions. A polynomial with complex coefficients of degree $n$ is an expression $P(z) = a_n z^n + \cdots + a_1 z + a_0$ , with $a_k$ complex and $a_n \neq 0$ ; a complex number $z_0$ is a root of $P$ when $P(z_0) = 0$ .

The proofs of the division algorithm, of the remainder theorem and of the factor theorem, given in the Course I lesson on the algebra of polynomials, use only the properties of addition, multiplication and division of the coefficients, which $\mathbb{C}$ shares with $\mathbb{R}$ according to the lesson on the complex numbers. Therefore they hold without change for polynomials with complex coefficients, and the same happens with the lemma on distinct roots and with the identity principle, from that same lesson (two polynomials that take the same value at every complex number have the same coefficients), and with the definition of the multiplicity of a root, from the lesson on the roots of higher-degree polynomials. What distinguishes $\mathbb{C}$ is the theorem stated below.

Admitted fact (fundamental theorem of algebra). Every polynomial of degree $n \geq 1$ with complex coefficients has at least one complex root.

As in the Course I lesson on the roots of polynomials, this theorem is accepted without proof: all its known proofs use tools that go beyond this course, such as the continuity of functions of two real variables or the study of functions of a complex variable. Carl Friedrich Gauss gave in 1799 the first almost complete proof, after the attempts of d’Alembert and of Euler. Its main consequence is that polynomial equations do not force us to enlarge $\mathbb{C}$ , as $x^2 = 2$ forced us to enlarge $\mathbb{Q}$ and $x^2 = -1$ to enlarge $\mathbb{R}$ : every complex polynomial splits completely in $\mathbb{C}$ .

TheoremFactorization over the complex numbers

Every polynomial $P$ of degree $n \geq 1$ with complex coefficients and leading coefficient $a_n$ is the product of $a_n$ and $n$ linear factors, as in the formula below, with $z_1, z_2, \ldots, z_n$ complex numbers that are not necessarily distinct. The roots of $P$ are exactly the numbers $z_k$ , and each one appears in the list as many times as its multiplicity indicates; therefore $P$ has exactly $n$ complex roots counted with multiplicity.

P(z) = a_n (z - z_1)(z - z_2) \cdots (z - z_n)

The proof proceeds by induction on the degree, as in the Course I lesson on summations, progressions and induction. The figure follows it in the complex plane with the polynomial $P(z) = z^5 + 2z^4 + 2z^3 - 4z^2 - 11z + 10$ , whose roots are $1$ , which is double, $-2$ , $-1 + 2i$ and $-1 - 2i$ .

Proof

$\resaltar{a_1 z + a_0} = \resaltar{a_1 \left( z + \frac{a_0}{a_1} \right)}$
Base case: a polynomial of degree $1$ is its leading coefficient times a single linear factor, whose root is $-\frac{a_0}{a_1}$ .
$P(z_1) = 0 \rightarrow P(z) = \resaltar{(z - z_1)} \dato{c}{\resaltar{C(z)}}, \qquad P(z) = (z - 1)\left( z^4 + 3z^3 + 5z^2 + z - 10 \right)$
Suppose the theorem true for the polynomials of degree $n$ , and let $P$ have degree $n + 1$ . By the fundamental theorem of algebra, $P$ has a root $z_1$ ; by the factor theorem, $P$ is the product of $z - z_1$ and a quotient $C$ of degree $n$ , whose leading coefficient is that of $P$ , as one sees by comparing the terms of highest degree. In the example, $z_1 = 1$ , and synthetic division gives the quotient in the formula.
$P(z) = (z - z_1) \resaltar{a_{n + 1} (z - z_2) \cdots (z - z_{n + 1})} = \dato{f}{\resaltar{a_{n + 1} (z - z_1)(z - z_2) \cdots (z - z_{n + 1})}}$
By the induction hypothesis, $C$ is its leading coefficient $a_{n + 1}$ times $n$ linear factors; on substituting, $P$ is written as desired. By the principle of induction, the factorization exists for every degree $n \geq 1$ .
$P(w) = a_n (w - z_1) \cdots (w - z_n) = 0 \leftrightarrow \resaltar{w = z_1 \vee \cdots \vee w = z_n}$
A product of complex numbers is zero only if one of its factors is, and $a_n \neq 0$ : a number $w$ is a root of $P$ if and only if it coincides with one of the $z_k$ . There are no roots other than those in the list.
$P(z) = a_n \resaltar{(z - r_1)^{k_1}} \cdots (z - r_s)^{k_s}, \qquad \resaltar{k_1 + k_2 + \cdots + k_s = n}$
We group equal factors: if $r_1, \ldots, r_s$ are the distinct roots and $r_j$ appears $k_j$ times, the quotient of $P$ by $(z - r_1)^{k_1}$ does not vanish at $r_1$ , because it is a product of nonzero factors; so $k_1$ is the multiplicity of $r_1$ , and the same happens with the others. The multiplicities add up to $n$ ; in the example, $2 + 1 + 1 + 1 = 5$ .

Since the multiplicity of each root is well defined, according to the Course I lesson on the roots of polynomials, the list $z_1, \ldots, z_n$ is determined by $P$ , up to order: the factorization is unique. From now on, the roots of a polynomial of degree $n$ mean that list of $n$ numbers, with the repetitions that the multiplicity indicates. If the coefficients are real, the nonreal roots come in conjugate pairs, by the conjugate roots theorem of the lesson on the complex numbers; multiplying the two factors of each pair gives a real quadratic factor, and so the factorization over the reals of the lesson on the roots of polynomials is recovered.

In the rest of this lesson

The relations between the roots and the coefficients
The power sums of the roots
Equations that reduce to simpler ones
Binomial and trinomial equations
Reciprocal equations
The cubic equation
The reduction to the depressed cubic
Cardano’s formula
The discriminant of the cubic
The irreducible case and the acceptance of complex numbers
The trigonometric solution
The quartic and Ferrari’s method
Solved problems
The real polynomial of least degree with two given roots
Power sums without finding the roots
Cardano’s formula with only one real root
Bombelli’s equation
A cubic with three real roots
Proposed problems
The reciprocals of the roots
The ladder leaning over a box
The waterline of a buoy
A sum of cube roots that is an integer
The discriminant and the differences of the roots