The real numbers leave many polynomial equations unsolved — x2+1=0 has no real answer, and neither does x4+4=0. Complex numbers fill this gap completely. The Fundamental Theorem of Algebra guarantees that every polynomial equation has solutions in C, and the number of solutions matches the polynomial's degree exactly. This completeness makes the complex field the natural setting for polynomial theory.
Polynomial Equations with Complex Solutions
Real numbers solve many polynomial equations but fail systematically on others. The equation x2−4=0 yields x=±2, both real. The equation x2+1=0 yields nothing — no real number squared equals −1. This gap plagued mathematics for centuries.
Complex numbers close the gap entirely. The equation x2+1=0 has two solutions in C: x=i and x=−i. Each satisfies i2=−1 and (−i)2=−1. The imaginary unit exists precisely to solve this equation.
Higher-degree equations with no real solutions also yield to complex numbers. Consider x4+4=0, equivalent to x4=−4. No real fourth power is negative, yet four complex solutions exist. Using De Moivre's theorem, we write −4=4cis(180°) and extract fourth roots: zk=2cis(45°+90°k) for k=0,1,2,3. The solutions are 1+i, −1+i, −1−i, and 1−i.
Even equations mixing real and complex solutions find all their roots in C. The equation x3+x=0 factors as x(x2+1)=0, giving x=0 (real) and x=±i (complex). Three roots total, matching the cubic degree.
The pattern holds universally: every polynomial equation of degree n has exactly n roots when we work in C and count roots according to their multiplicity.
The Fundamental Theorem of Algebra
The cornerstone of polynomial theory makes a sweeping guarantee: every non-constant polynomial with complex coefficients has at least one complex root. This is the Fundamental Theorem of Algebra, first rigorously proved by Carl Friedrich Gauss in 1799.
The statement seems modest — just one root guaranteed. But its consequences are profound. If polynomial p(z) of degree n≥1 has a root z1, then p(z)=(z−z1)q(z) where q(z) has degree n−1. If n−1≥1, the theorem applies again: q(z) has a root z2, so q(z)=(z−z2)r(z). Continue until only a constant remains.
The conclusion: every polynomial of degree n factors completely as:
p(z)=an(z−z1)(z−z2)⋯(z−zn)
where an is the leading coefficient and z1,z2,…,zn are the n roots (possibly with repetitions). A polynomial of degree n has exactly n roots in C, counted with multiplicity.
This completeness property means C is algebraically closed. Every polynomial equation solvable anywhere is solvable in C. No further extension of the number system is needed to find polynomial roots — complex numbers are the final destination.
The theorem's proof lies beyond elementary methods, requiring tools from analysis or topology. But its implications pervade all of algebra: we can always factor polynomials, always find roots, always reduce polynomial equations to products of linear factors over C.
Factoring Polynomials over C
The Fundamental Theorem guarantees that every polynomial splits into linear factors over C. The factorization reveals the polynomial's roots explicitly and connects algebraic structure to geometric information in the complex plane.
For a polynomial p(z)=anzn+an−1zn−1+⋯+a1z+a0, the complete factorization takes the form:
p(z)=an(z−z1)(z−z2)⋯(z−zn)
The roots z1,z2,…,zn may repeat. A root appearing m times is said to have multiplicity m. The polynomial p(z)=(z−2)3(z+i)(z−i) has degree 5, with root 2 of multiplicity 3 and simple roots ±i.
The factorization of z3−1 illustrates the interplay with roots of unity. The roots satisfy z3=1, giving z0=1, z1=cis(120°)=−21+23i, and z2=cis(240°)=−21−23i. Writing ω=cis(120°):
z3−1=(z−1)(z−ω)(z−ω2)
Multiplying the last two factors produces a real quadratic: (z−ω)(z−ω2)=z2+z+1. Thus z3−1=(z−1)(z2+z+1), the familiar factorization over R. Over C, we factor further into linear terms.
Every polynomial admits such complete factorization. Real polynomials may resist factoring over R into linear terms, but over C, every polynomial yields completely.
Vieta's Formulas
A remarkable correspondence links a polynomial's coefficients to symmetric combinations of its roots. These relationships, known as Vieta's formulas after François Viète, allow information to flow between the algebraic expression of a polynomial and the location of its roots.
For a monic quadratic z2+bz+c=0 with roots z1 and z2:
z1+z2=−bz1⋅z2=c
The sum of roots equals the negative of the linear coefficient. The product of roots equals the constant term. Given roots 3 and −5, the polynomial is z2−(−2)z+(−15)=z2+2z−15.
For a monic cubic z3+bz2+cz+d=0 with roots z1, z2, z3:
z1+z2+z3=−b
z1z2+z1z3+z2z3=c
z1z2z3=−d
The pattern continues for higher degrees. The k-th symmetric sum of roots (all products of k roots) equals (−1)k times the coefficient of zn−k in a monic polynomial of degree n.
These formulas prove useful in both directions. Given a polynomial, we can compute sums and products of roots without finding the roots explicitly. Given information about roots, we can reconstruct polynomial coefficients. Problems asking for the sum of squares of roots or similar expressions often yield to Vieta's formulas combined with algebraic identities like (z1+z2)2=z12+2z1z2+z22.
Polynomials with Real Coefficients
Polynomials whose coefficients are all real numbers exhibit special structure: their non-real roots always come in conjugate pairs. If z0 is a root, so is z0ˉ.
The proof exploits how conjugation interacts with polynomial evaluation. Let p(z)=anzn+an−1zn−1+⋯+a0 with all ak real. For any complex z:
The key step uses ak=ak since each coefficient is real, and the property zk=(zˉ)k.
Now suppose p(z0)=0. Then p(z0)=0ˉ=0, so p(z0ˉ)=0. The conjugate is also a root.
Consequences abound. A real quadratic with complex roots has the form (z−z0)(z−z0ˉ) with conjugate pair z0,z0ˉ. This product expands to z2−2Re(z0)z+∣z0∣2, a real quadratic as expected.
A real polynomial of odd degree must have at least one real root. Complex roots pair off, consuming an even number of the n roots. With n odd, at least one root remains unpaired — and an unpaired root of a real polynomial must be real (otherwise its conjugate would also be a root).
This explains why every real cubic crosses the real axis: it must have either three real roots or one real root plus a conjugate pair.
Quadratic Equations with Complex Coefficients
The quadratic formula extends unchanged into the complex domain. For the equation az2+bz+c=0 with complex coefficients a, b, c (and a=0):
z=2a−b±b2−4ac
The formula remains valid, but a new challenge emerges: the discriminant Δ=b2−4ac may itself be complex, requiring us to compute the square root of a complex number.
Consider z2+2iz−1=0. Here a=1, b=2i, c=−1. The discriminant is:
Δ=(2i)2−4(1)(−1)=−4+4=0
With discriminant zero, one repeated root exists: z=2−2i=−i.
For a case with complex discriminant, take z2+z+(1−i)=0. The discriminant is:
Δ=1−4(1−i)=1−4+4i=−3+4i
Finding −3+4i requires the technique from De Moivre's theorem. Convert −3+4i to trigonometric form: modulus 9+16=5, argument θ=arctan(4/(−3)) adjusted to the second quadrant. The square roots have modulus 5 and arguments θ/2 and θ/2+180°.
The process is lengthier than real quadratics but entirely systematic. Every quadratic with complex coefficients has two roots in C (counting multiplicity), accessible through the same formula that handles real coefficients.
Solving zn=w
Equations of the form zn=w, where w is a given complex number, have exactly n solutions. The technique applies De Moivre's theorem systematically.
The procedure:
1. Express w in trigonometric form: w=Rcisϕ 2. Apply the root formula: zk=R1/ncis(nϕ+360°k) for k=0,1,…,n−1 3. Convert to algebraic form if needed
Example: Solve z4=−16.
First write −16 in trigonometric form. The modulus is 16, and the argument is 180° (negative real axis). So −16=16cis(180°).
The fourth roots have modulus 161/4=2 and arguments 4180°+360°k=45°+90°k:
z0=2cis(45°)=2⋅22(1+i)=2(1+i)
z1=2cis(135°)=2⋅22(−1+i)=2(−1+i)
z2=2cis(225°)=2⋅22(−1−i)=2(−1−i)
z3=2cis(315°)=2⋅22(1−i)=2(1−i)
Four roots forming a square on a circle of radius 2, rotated 45° from the axes. Each satisfies (zk)4=−16.
Solving zn=zˉ
The equation zn=zˉ combines exponentiation with conjugation, producing a distinctive solution set. Unlike zn=w which has exactly n roots, this equation has n+2 solutions.
Begin by writing z=reiθ in exponential form. Then zn=rneinθ and zˉ=re−iθ. The equation becomes:
rneinθ=re−iθ
Two conditions emerge from matching moduli and arguments.
Moduli: rn=r, giving rn−r=r(rn−1−1)=0. Either r=0 or rn−1=1.
Arguments: nθ=−θ+360°k for integer k, so (n+1)θ=360°k, giving θ=n+1360°k.
The solution r=0 yields z=0.
The solutions with rn−1=1 require r=1 (since r≥0). These solutions lie on the unit circle with arguments θ=n+1360°k for k=0,1,…,n.
Counting: one solution at the origin, plus n+1 solutions on the unit circle, totaling n+2 solutions.
Example: z2=zˉ has 2+2=4 solutions.
From r2=r: r=0 or r=1. From 3θ=360°k: θ=0°,120°,240°.
Equations relating z and zˉ often describe geometric loci in the complex plane. The conjugate identities translate these equations into conditions on real and imaginary parts.
Type 1: $z + \bar{z} = k$ for real constant $k$.
Since z+zˉ=2Re(z), the equation becomes 2a=k, or a=k/2. The solution set is the vertical line where the real part equals k/2. Every point on this line satisfies the equation; no point off the line does.
Type 2: $z - \bar{z} = ki$ for real constant $k$.
Since z−zˉ=2bi, the equation becomes 2bi=ki, so b=k/2. The solution set is the horizontal line where the imaginary part equals k/2.
Type 3: $z \cdot \bar{z} = k$ for positive real constant $k$.
Since z⋅zˉ=∣z∣2, the equation becomes ∣z∣2=k, or ∣z∣=k. The solution set is the circle centered at the origin with radius k.
Combining conditions produces intersections of these loci. The system:
z+zˉ=4andz⋅zˉ=5
The first equation restricts to the vertical line Re(z)=2. The second restricts to the circle ∣z∣=5. Their intersection: points with real part 2 on the circle of radius 5. Writing z=2+bi, we need 4+b2=5, so b=±1. Solutions: z=2+i and z=2−i.
This geometric approach often simplifies what algebraic manipulation would make tedious.
Applications
Complex roots of polynomials appear throughout science and engineering, carrying meaning far beyond pure mathematics. The location of roots in the complex plane determines the behavior of physical and computational systems.
Signal Processing: The transfer function of a linear filter is a ratio of polynomials in a complex variable. The filter's frequency response — which frequencies pass through and which are attenuated — depends directly on where the polynomial roots lie. Roots near the unit circle create sharp peaks or nulls in the response. Filter design amounts to placing roots strategically in the complex plane.
Control Theory: A dynamical system's stability hinges on the roots of its characteristic polynomial. If all roots have negative real parts (lying in the left half-plane), the system is stable — disturbances decay over time. Roots with positive real parts indicate instability — small perturbations grow without bound. Control engineers reshape root locations through feedback to achieve desired stability margins.
Geometry: Regular polygons emerge as roots of unity. The n-th roots of unity form the vertices of a regular n-gon inscribed in the unit circle. Solving zn=1 constructs the polygon automatically. More generally, the roots of zn=w form a regular n-gon of radius ∣w∣1/n.
Number Theory: Cyclotomic polynomials — the minimal polynomials whose roots are primitive roots of unity — encode deep arithmetic structure. Their coefficients, degrees, and factorization properties connect to prime numbers, Galois theory, and algebraic integers.
The Fundamental Theorem of Algebra ensures that these applications never encounter polynomials without roots. In C, every polynomial equation has solutions, making complex numbers the natural language for polynomial-based analysis across all fields.