Complex numbers in polar form: synthesis and challenges

Summary

This lesson closes the unit devoted to complex numbers in polar form. It gathers its definitions and theorems into a single deductive chain, which starts from the product theorem in polar form and arrives, through De Moivre’s formula, the $n$ th roots, the complex exponential and the geometry of the plane, at the factorization of polynomials and at the formulas of Cardano and Ferrari. It points out what each link owes to the course Algebra and Geometry I and to the two earlier units of this one, and how the following unit, on matrices and algebraic structures, reinterprets the same facts. It ends with ten challenges, harder than the proposed problems of the unit, which combine several of its lessons.

Learning objectives

Reconstruct the deductive chain of the unit, from the product theorem in polar form to the formulas of Cardano and Ferrari, indicating which result each theorem depends on.
Relate the results of the unit to those of Course I (induction, the binomial theorem, the greatest common divisor, polynomials) and to those of the earlier units (polar coordinates and trigonometric formulas), and anticipate their matrix and structural reading in the following unit.
Solve problems that require combining the polar form, the roots of unity, the complex exponential, geometry with complex numbers and Vieta’s formulas.

What the unit achieved

The unit started from a gap: in the Course I lesson on the complex numbers, the product of two complex numbers could be computed, but not understood. The lesson on the polar form wrote each nonzero complex number as $r\left( \cos \theta + i \sin \theta \right)$ , proved that this expression is unique up to full turns and, with the sum and difference formulas, proved the theorem on which everything else rests: in a product, the moduli are multiplied and the arguments are added. From it followed that multiplying by a complex number is rotating and dilating, that the conjugate and the inverse have the opposite argument, and that the principal argument of a product coincides with the sum of the principal arguments only up to a full turn.

If the product adds arguments, the power multiplies them. The lesson on De Moivre’s formula proved this by induction for every integer exponent and, comparing the formula with the binomial theorem, obtained the multiple-angle formulas; with the lemma $z^n + z^{-n} = 2\cos(n\theta)$ it linearized the powers of the sine and the cosine, with the geometric sum of complex ratio it summed sines and cosines, and with the power $(1 + i)^n$ , the binomial coefficients of even index and those of odd index with alternating signs. Now, if raising to the power $n$ multiplies the angle by $n$ , extracting a root divides it: the lesson on the $n$ th roots proved that every nonzero complex number has exactly $n$ $n$ th roots, the vertices of a regular polygon, and that the roots of unity are the powers of a single one, add up to zero, factor $z^n - 1$ and are primitive exactly when their index is coprime to $n$ .

The lesson on the complex exponential observed that all these rules are the laws of exponents and, with the series of $e^x$ , of the cosine and of the sine, which it admitted without proof because they require differential calculus, showed that the sum of the exponential series at $i\theta$ is $\cos \theta + i \sin \theta$ . It therefore defined $e^{i\theta} = \cos \theta + i \sin \theta$ and $e^{a + bi} = e^a e^{ib}$ , proved that the exponential turns sums into products and is periodic with period $2\pi i$ , expressed the cosine and the sine by means of exponentials, decided which trigonometric identities are preserved when they are extended to the complex numbers, discovered the hyperbolic functions in them and, by inverting the exponential, obtained a logarithm with infinitely many values and powers with a complex exponent. With this notation, the lesson on geometry with complex numbers wrote rotations as $c + e^{i\varphi}(z - c)$ and reflections as $e^{2i\alpha}\conj{z}$ , read collinearity and perpendicularity in a quotient, defined the orientation of a triangle by the sign of the imaginary part of that quotient and proved that it does not depend on the vertex one starts from, that translations and rotations preserve it and that reflections reverse it, characterized the equilateral triangle with the cube root of unity, and proved the theorem on the circle of Apollonius and Napoleon’s theorem.

Finally, the lesson on polynomial equations brought everything together on the ground where complex numbers were born. From the fundamental theorem of algebra, which is admitted without proof, it deduced that every polynomial of degree $n$ is a product of $n$ linear factors, proved Vieta’s formulas and solved binomial, trinomial and reciprocal equations; Cardano’s formula required the cube roots from the lesson on $n$ th roots, the discriminant classified the roots of a real cubic, the irreducible case was solved with the triple-angle formula, and Ferrari’s method reduced the quartic to a cubic; Hölder’s theorem and the Abel-Ruffini theorem, which are admitted, explained why the detour through the complex numbers is unavoidable in the irreducible case and why there are no similar formulas from degree five on. Hence the complete chain is as follows: product theorem, De Moivre’s formula, $n$ th roots, complex exponential, geometry of the plane and polynomial equations, so that each link is, in essence, a new reading of the first.

It is worth listing exactly what the unit admits without proof, because it is little and well localized. From Euclidean geometry, the facts gathered in the Course I lesson on the equation of a line, of which the unit only needs the notion of similarity and the fact that equal angles entail proportional sides, in the theorem on the similar triangles of the product. From differential calculus, the series of $e^x$ , of the cosine and of the sine, which the lesson on the complex exponential uses to motivate Euler’s formula and which are proved in the course Calculus I. From algebra, three results stated in the lesson on polynomial equations: the fundamental theorem of algebra, on which the complete factorization depends, and Hölder’s theorem and the Abel-Ruffini theorem, on which no link rests and which only explain why the irreducible case requires the detour through the complex numbers and why there are no formulas by radicals from degree five on. Finally, the example of the spiral of powers, in the lesson on De Moivre’s formula, mentions without using it the property of the logarithmic spiral of cutting every line through the origin at the same angle, which is also proved with differential calculus. Everything else is proved in the unit, in the two preceding ones or in Course I; in the chain, the Euclidean facts and the fundamental theorem appear as admitted supports, and the series, on which no link rests because the imaginary exponential is adopted as a definition, do not.

The following figure gathers the four segments of the unit into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again: the theorem on equality in polar form, which is established in the first link, and the results of earlier units or of Course I, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AEquality in polar form. Two complex numbers $r_1\left( \cos \theta_1 + i \sin \theta_1 \right)$ and $r_2\left( \cos \theta_2 + i \sin \theta_2 \right)$ , with $r_1$ and $r_2$ positive, are equal if and only if $r_1 = r_2$ and $\theta_1 - \theta_2$ is an integer multiple of $2\pi$ . Established in: 1. Equality in polar form.
BTrigonometric formulas. The Pythagorean identity, the cosine and the sine of a sum and of a difference, the double angle, the half angle and the triple angle, the parity and the periodicity of the sine and the cosine, their zeros, the characterization of coterminal angles, the monotonicity of the cosine on $\icc{0}{\pi}$ and the law of cosines, proved in units 1 and 2, on the trigonometric functions and on analytic trigonometry. Admitted without proof in this unit.
CEuclidean geometry. In the Course I unit on the analytic geometry of the plane, the facts of Euclidean geometry are admitted: the Pythagorean theorem and its converse, parallels, the intercept theorem and the similarity of triangles, the sum of the angles of a triangle and the isosceles triangle. Admitted without proof in this unit.
DPrinciple of induction. If a property holds for the first natural number and passes from each natural number to its successor, it holds for all of them; proved in the Course I unit on the natural numbers. Admitted without proof in this unit.
EBinomial theorem and geometric sum. For every natural number $n$ we have $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n - k} y^k$ , and if $q \neq 1$ , $1 + q + \cdots + q^{n - 1} = \frac{q^n - 1}{q - 1}$ ; proved in the Course I unit on the natural numbers, both computations use only the properties of addition and multiplication and also hold with complex numbers. Admitted without proof in this unit.
FDivision with remainder and Bézout. Every integer can be written as $qn + r$ with $0 \leq r \leq n - 1$ , and if $\operatorname{gcd}(k, n) = 1$ there exist integers $x$ and $y$ such that $kx + ny = 1$ ; proved in the Course I unit on the natural numbers. Admitted without proof in this unit.
GReal exponential. The real exponential is positive, injective and bijective onto the positive reals, satisfies $e^a e^c = e^{a + c}$ and defines the hyperbolic cosine and sine; proved in the Course I unit on the exponential and the logarithm. Admitted without proof in this unit.
HFactor theorem and fundamental theorem. If $z_0$ is a root of the polynomial $P$ , then $z - z_0$ divides $P$ , which was proved in the Course I unit on algebraic functions and holds over the complex numbers; and every complex polynomial of degree $n \geq 1$ has some complex root, which is admitted without proof. Admitted without proof in this unit.

Segment 1 The polar form and the product
- Write every nonzero complex number in polar form and prove that this expression is unique up to full turns.
- Prove with the sum and difference formulas that, in a product, the moduli are multiplied and the arguments are added.
- Deduce from this the conjugate and the inverse in polar form, and the behavior of the principal argument of a product.
- Interpret multiplication by a complex number as a rotation composed with a dilation.
1. Equality in polar form. An argument of $z = a + bi \neq 0$ is any real $\theta$ with $a = \abs{z} \cos \theta$ and $b = \abs{z} \sin \theta$ ; the complex number $r\left( \cos \theta + i \sin \theta \right)$ , with $r > 0$ , has modulus $r$ and argument $\theta$ , and two such expressions are equal if and only if they have the same modulus and their arguments differ by an integer multiple of $2\pi$ .
  Rests on: B (Trigonometric formulas). Proved in: The polar form of complex numbers.
2. Product and quotient in polar form. If $z$ has modulus $r$ and argument $\theta$ , and $w$ has modulus $s$ and argument $\varphi$ , the product $zw$ has modulus $rs$ and argument $\theta + \varphi$ , and the quotient $\frac{z}{w}$ has modulus $\frac{r}{s}$ and argument $\theta - \varphi$ .
  Rests on: A (Equality in polar form), B (Trigonometric formulas). Proved in: The polar form of complex numbers.
3. Conjugate and inverse. If $z = r\left( \cos \theta + i \sin \theta \right)$ , with $r > 0$ , the conjugate of $z$ has modulus $r$ and argument $-\theta$ , and the inverse $\frac{1}{z}$ has modulus $\frac{1}{r}$ and argument $-\theta$ .
  Uses: 2. Product and quotient in polar form. Rests on: A (Equality in polar form), B (Trigonometric formulas). Proved in: The polar form of complex numbers.
4. Principal argument of a product. The principal argument of $zw$ is $\operatorname{Arg} z + \operatorname{Arg} w + 2k\pi$ , with $k$ equal to $-1$ , $0$ or $1$ , and $k = 0$ exactly when that sum lies in $\ioc{-\pi}{\pi}$ .
  Uses: 2. Product and quotient in polar form. Rests on: A (Equality in polar form). Proved in: The polar form of complex numbers.
5. Similar triangles of the product. If $z \neq 0$ and $w$ is not real, the triangle with vertices $0$ , $z$ and $zw$ is similar to the one with vertices $0$ , $1$ and $w$ , with ratio $\abs{z}$ : multiplying by $z$ is rotating and dilating.
  Uses: 2. Product and quotient in polar form. Rests on: C (Euclidean geometry), B (Trigonometric formulas). Proved in: The polar form of complex numbers.
Segment 2 Powers and roots
- Prove by induction De Moivre’s formula for every integer exponent.
- Obtain from it and from the binomial theorem the multiple-angle formulas, linearize the powers of the cosine and sum sines and cosines with a geometric sum.
- Prove that every nonzero complex number has exactly $n$ $n$ th roots, the vertices of a regular polygon.
- Study the roots of unity: their powers, their sum, the factorization of $z^n - 1$ and the primitive roots.
1. De Moivre’s formula. For every real $\theta$ and every integer $n$ we have $\left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta)$ ; therefore, if $z = r\left( \cos \theta + i \sin \theta \right)$ , the power $z^n$ has modulus $r^n$ and argument $n\theta$ .
  Uses: 2. Product and quotient in polar form, 3. Conjugate and inverse. Rests on: A (Equality in polar form), D (Principle of induction), B (Trigonometric formulas). Proved in: De Moivre’s formula and powers of complex numbers.
2. Multiple angles. The cosine of $n\theta$ is the alternating sum of the terms $\binom{n}{k} \cos^{n - k} \theta \sin^k \theta$ with even index $k$ , and the sine of $n\theta$ , that of the terms with odd index.
  Uses: 6. De Moivre’s formula. Rests on: E (Binomial theorem and geometric sum). Proved in: De Moivre’s formula and powers of complex numbers.
3. Linearization. If $z = \cos \theta + i \sin \theta$ , we have $z^n + z^{-n} = 2 \cos(n\theta)$ and $z^n - z^{-n} = 2i \sin(n\theta)$ ; from this follow, for example, $\cos^3 \theta = \frac{\cos 3\theta + 3 \cos \theta}{4}$ and $\cos^4 \theta = \frac{\cos 4\theta + 4 \cos 2\theta + 3}{8}$ .
  Uses: 6. De Moivre’s formula. Rests on: E (Binomial theorem and geometric sum), B (Trigonometric formulas). Proved in: De Moivre’s formula and powers of complex numbers.
4. Sums of sines and cosines. If $\theta$ is not an integer multiple of $2\pi$ , the sum of the complex numbers $\cos(k\theta) + i \sin(k\theta)$ , for $k$ from $0$ to $n - 1$ , is the real number $\frac{\sin \frac{n\theta}{2}}{\sin \frac{\theta}{2}}$ times the complex number of modulus $1$ and argument $\frac{(n - 1)\theta}{2}$ , which gives the sums of the cosines and of the sines.
  Uses: 2. Product and quotient in polar form, 6. De Moivre’s formula. Rests on: A (Equality in polar form), E (Binomial theorem and geometric sum), B (Trigonometric formulas). Proved in: De Moivre’s formula and powers of complex numbers.
5. nth roots. If $w = R\left( \cos \varphi + i \sin \varphi \right) \neq 0$ , the equation $z^n = w$ has exactly $n$ solutions, with modulus $\sqrt[n]{R}$ and arguments $\frac{\varphi + 2k\pi}{n}$ with $0 \leq k \leq n - 1$ , which for $n \geq 3$ are the vertices of a regular polygon inscribed in the circle of radius $\sqrt[n]{R}$ .
  Uses: 2. Product and quotient in polar form, 6. De Moivre’s formula. Rests on: A (Equality in polar form), F (Division with remainder and Bézout), B (Trigonometric formulas), D (Principle of induction). Proved in: The nth roots of a complex number and the roots of unity.
6. Roots of unity as powers. The $n$ th roots of unity are the powers $\omega_k = \omega^k$ of $\omega = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}$ ; they are multiplied by adding the indices modulo $n$ , the inverse of each one is its conjugate, and multiplying by one of them permutes them.
  Uses: 3. Conjugate and inverse, 6. De Moivre’s formula, 10. nth roots. Rests on: F (Division with remainder and Bézout). Proved in: The nth roots of a complex number and the roots of unity.
7. Sum of the powers of the roots. If $n \geq 2$ , the sum of the $m$ th powers of the $n$ $n$ th roots of unity equals $n$ if $n$ divides $m$ , and $0$ otherwise; in particular, the roots add up to zero.
  Uses: 11. Roots of unity as powers. Rests on: A (Equality in polar form), E (Binomial theorem and geometric sum). Proved in: The nth roots of a complex number and the roots of unity.
8. Factorization of z to the n minus 1. For every natural number $n$ , the polynomial $z^n - 1$ is the product of the $n$ factors $z - \omega_k$ , with $0 \leq k \leq n - 1$ .
  Uses: 10. nth roots. Rests on: H (Factor theorem and fundamental theorem). Proved in: The nth roots of a complex number and the roots of unity.
9. Primitive roots. The root $\omega_k$ , with $1 \leq k \leq n$ , is primitive, that is, its powers give all the $n$ th roots of unity, if and only if $\operatorname{gcd}(k, n) = 1$ .
  Uses: 11. Roots of unity as powers, 12. Sum of the powers of the roots. Rests on: F (Division with remainder and Bézout). Proved in: The nth roots of a complex number and the roots of unity.
10. Real factorization of x to the n minus 1. Each pair of conjugate nonreal roots $\omega_k$ and $\omega_{n - k}$ gives the irreducible factor $x^2 - 2x \cos \frac{2k\pi}{n} + 1$ , so that $x^n - 1$ is the product of $x - 1$ , of $x + 1$ if $n$ is even, and of those quadratic factors.
  Uses: 3. Conjugate and inverse, 11. Roots of unity as powers, 13. Factorization of z to the n minus 1. Rests on: A (Equality in polar form), B (Trigonometric formulas). Proved in: The nth roots of a complex number and the roots of unity.
Segment 3 The complex exponential and geometry
- Define $e^{i\theta}$ by Euler’s formula and $e^{a + bi} = e^a e^{ib}$ , and recognize the laws of exponents in the earlier rules.
- Prove that the exponential turns sums into products and is periodic with period $2\pi i$ .
- Express the cosine and the sine by means of exponentials, relate them to the hyperbolic functions and obtain the logarithms of a complex number.
- Write the rotations and the reflections of the plane with complex numbers, read collinearity and perpendicularity in a quotient, define the orientation of a triangle and prove its invariance, and characterize the equilateral triangle.
1. Imaginary exponential. The complex number $e^{i\theta} = \cos \theta + i \sin \theta$ has modulus $1$ , and we have $e^{i\alpha} e^{i\beta} = e^{i(\alpha + \beta)}$ , $\left( e^{i\theta} \right)^n = e^{in\theta}$ and $\left( e^{i\theta} \right)^{-1} = e^{-i\theta}$ , which is its conjugate; moreover, $e^{i\alpha} = e^{i\beta}$ if and only if $\alpha - \beta$ is an integer multiple of $2\pi$ .
  Uses: 2. Product and quotient in polar form, 3. Conjugate and inverse, 6. De Moivre’s formula. Rests on: A (Equality in polar form), B (Trigonometric formulas). Proved in: The complex exponential and Euler’s formula.
2. Exponential of a sum. If $z = a + bi$ and $e^z = e^a e^{ib}$ , the modulus of $e^z$ is $e^a$ , $b$ is one of its arguments and $e^z$ is never zero; and for all complex numbers $z$ and $w$ we have $e^{z + w} = e^z e^w$ .
  Uses: 2. Product and quotient in polar form, 16. Imaginary exponential. Rests on: A (Equality in polar form), G (Real exponential). Proved in: The complex exponential and Euler’s formula.
3. Periodicity of the exponential. We have $e^z = 1$ if and only if $z = 2k\pi i$ for some integer $k$ ; therefore, $e^z = e^w$ if and only if $z - w$ is an integer multiple of $2\pi i$ .
  Uses: 17. Exponential of a sum. Rests on: A (Equality in polar form), G (Real exponential). Proved in: The complex exponential and Euler’s formula.
4. Euler’s formulas. We have $\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$ ; with them the cosine and the sine of a complex number are defined, which preserve the addition, double-angle and power-reduction formulas, but not the half-angle formulas with a double sign, and one obtains $\cos(ix) = \cosh x$ and $\sin(ix) = i \sinh x$ .
  Uses: 16. Imaginary exponential, 17. Exponential of a sum. Rests on: B (Trigonometric formulas), G (Real exponential). Proved in: The complex exponential and Euler’s formula.
5. Logarithms of a complex number. If $z \neq 0$ , the solutions of $e^w = z$ are $w = \ln \abs{z} + i\left( \operatorname{Arg} z + 2k\pi \right)$ , with $k$ an integer; and the values of $z^{\frac{1}{n}}$ defined with them are exactly the $n$ $n$ th roots of $z$ .
  Uses: 10. nth roots, 11. Roots of unity as powers, 17. Exponential of a sum, 18. Periodicity of the exponential. Rests on: A (Equality in polar form), G (Real exponential). Proved in: The complex exponential and Euler’s formula.
6. Rotation about a point. The transformation $R(z) = c + e^{i\varphi}(z - c)$ leaves $c$ fixed, preserves distances and rotates through an angle $\varphi$ the line segment joining each point with $c$ : it is the rotation about $c$ through the angle $\varphi$ .
  Uses: 2. Product and quotient in polar form, 16. Imaginary exponential. Proved in: Geometry with complex numbers.
7. Reflections and their composition. The reflection in the line through the origin with angle $\alpha$ is the isometry $S(z) = e^{2i\alpha} \conj{z}$ , and the composition of the reflections with angles $\alpha$ and $\beta$ is the rotation about the origin through the angle $2(\beta - \alpha)$ .
  Uses: 3. Conjugate and inverse, 16. Imaginary exponential, 21. Rotation about a point. Proved in: Geometry with complex numbers.
8. Collinearity and orientation. If $q = \frac{z_3 - z_1}{z_2 - z_1}$ , the oriented angle from the line segment from $z_1$ to $z_2$ to the one from $z_1$ to $z_3$ is an argument of $q$ ; the points are collinear if and only if $q$ is real, and the line segments are perpendicular if and only if $q$ is purely imaginary. If they are not collinear, the triangle $z_1 z_2 z_3$ is positively oriented when $\operatorname{Im} q > 0$ ; the orientation does not depend on the initial vertex, it is reversed by swapping two vertices and by applying a reflection, and it is preserved by rotations.
  Uses: 2. Product and quotient in polar form, 16. Imaginary exponential, 21. Rotation about a point, 22. Reflections and their composition. Proved in: Geometry with complex numbers.
9. Equilateral triangle. If $z_1 \neq z_2$ , the points $z_1$ , $z_2$ and $z_3$ are, in that order, the vertices of a positively oriented equilateral triangle if and only if $z_1 + \omega z_2 + \omega^2 z_3 = 0$ , with $\omega = e^{\frac{2\pi i}{3}}$ ; the proof reduces the condition to $\abs{q} = \abs{q - 1} = 1$ with $\operatorname{Im} q > 0$ , that is, to $q = e^{\frac{i\pi}{3}}$ .
  Uses: 16. Imaginary exponential, 12. Sum of the powers of the roots, 23. Collinearity and orientation. Proved in: Geometry with complex numbers.
Segment 4 Polynomial equations
- Deduce from the fundamental theorem of algebra that every complex polynomial of degree $n$ is a product of $n$ linear factors.
- Prove Vieta’s formulas relating the roots and the coefficients.
- Solve the cubic with Cardano’s formula, classify its real roots with the discriminant and solve the irreducible case with the triple angle.
- Reduce the quartic to a cubic with Ferrari’s method.
1. Factorization over the complex numbers. Every complex polynomial of degree $n \geq 1$ and leading coefficient $a_n$ is $a_n (z - z_1) \cdots (z - z_n)$ , and it has exactly $n$ roots counted with multiplicity.
  Rests on: D (Principle of induction), H (Factor theorem and fundamental theorem). Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.
2. Vieta’s formulas. The elementary symmetric functions of the roots satisfy $\sigma_k = (-1)^k \frac{a_{n - k}}{a_n}$ ; in particular, the sum of the roots is $-\frac{a_{n - 1}}{a_n}$ and their product is $(-1)^n \frac{a_0}{a_n}$ .
  Uses: 25. Factorization over the complex numbers. Rests on: D (Principle of induction). Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.
3. Cardano’s formula. The substitution $x = t - \frac{b}{3a}$ reduces the cubic to $t^3 + pt + q = 0$ ; if $u$ is a cube root of a root $U$ of the resolvent quadratic $s^2 + qs - \frac{p^3}{27} = 0$ and $v = -\frac{p}{3u}$ , its roots are $u + v$ , $\omega u + \omega^2 v$ and $\omega^2 u + \omega v$ .
  Uses: 10. nth roots, 11. Roots of unity as powers, 12. Sum of the powers of the roots, 26. Vieta’s formulas. Rests on: E (Binomial theorem and geometric sum). Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.
4. Discriminant of the cubic. If $p$ and $q$ are real and $\Delta = -\left( 4p^3 + 27q^2 \right)$ , the cubic $t^3 + pt + q = 0$ has one real root and two conjugate nonreal roots if $\Delta < 0$ , real roots with some multiple root if $\Delta = 0$ , and three distinct real roots if $\Delta > 0$ .
  Uses: 3. Conjugate and inverse, 11. Roots of unity as powers, 27. Cardano’s formula. Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.
5. Trigonometric solution. If $\Delta > 0$ , with $m = 2\sqrt{-\frac{p}{3}}$ and $c = \frac{3q}{pm}$ in $\ioo{-1}{1}$ , the three real roots are $m \cos \frac{\arccos c + 2k\pi}{3}$ , for $k = 0, 1, 2$ .
  Uses: 25. Factorization over the complex numbers, 28. Discriminant of the cubic. Rests on: B (Trigonometric formulas). Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.
6. Ferrari’s method. The substitution $x = y - \frac{b}{4a}$ reduces the quartic to $y^4 + py^2 + qy + r = 0$ ; if $q \neq 0$ and $m$ is a root of its resolvent cubic, with $s^2 = 2m$ , the quartic is the product of two quadratic factors whose roots are its own.
  Uses: 10. nth roots, 25. Factorization over the complex numbers, 27. Cardano’s formula. Rests on: E (Binomial theorem and geometric sum). Proved in: Complex polynomial equations: the Cardano and Ferrari formulas.

Toward matrices and algebraic structures. The chain ends where complex numbers become structure: the product by $e^{i\varphi}$ will be a rotation matrix, the squared modulus a determinant, and the roots of unity, which multiply as the remainders of division by $n$ add, the first example of a cyclic group.