What the coefficients already know

Every polynomial of degree n has exactly n roots, and the proof of that says nothing whatever about where they are. It watches a loop wind round the origin, notices that the winding count has to jump somewhere, and concludes that the curve passed through zero — an argument that would work just as well if the roots were somewhere else entirely.

That leaves an awkward gap. The roots exist, they are usually not writable in any useful closed form, and yet a great deal about them can be read straight off the coefficients without a moment of searching.

Two numbers are visible in that picture and neither needed the roots. The average of the roots is minus the second coefficient divided by the leading one; their product is the constant term divided by the leading one, with a sign depending on the degree. Both are exact, both are arithmetic on the coefficients, and both hold for polynomials nobody can solve.

Multiplying the factors out #

The relations come from doing the obvious thing in the obvious direction and then reading the result backwards.

A polynomial with roots r₁, r₂ and r₃ and leading coefficient 1 is the product (x − r₁)(x − r₂)(x − r₃). Multiplying that out gives

$$x^3 - (r_1 + r_2 + r_3)x^2 + (r_1r_2 + r_1r_3 + r_2r_3)x - r_1r_2r_3.$$

Every coefficient of the expanded form is a combination of the roots, and the combinations are the ones that treat all the roots alike: the sum, the sum of pairwise products, the product. Comparing with a polynomial written as x³ + bx² + cx + d gives three identities at a stroke — the sum of the roots is −b, the sum of the products in pairs is c, and the product is −d.

That is Vieta’s relations, and the derivation is a single expansion. Nothing subtle happens; what makes it worth a name is what it says when read in the other direction. A polynomial’s coefficients are not arbitrary numbers attached to powers of x. They are the symmetric functions of the roots, and the polynomial is a compressed way of writing that list.

Why only the symmetric combinations #

There is a reason the sum and the product appear and no individual root does, and it is worth pinning down because it is the beginning of a much larger subject.

The roots of a polynomial have no order. Nothing about x³ − 6x² + 11x − 6 picks out which of 1, 2 and 3 is to be called the first root; that labelling was invented when they were written down. So any quantity computed from the coefficients alone must be unchanged when the roots are relabelled — it cannot depend on a choice that was never made.

The combinations of several numbers that survive every relabelling are the symmetric ones, and the fundamental theorem of symmetric polynomials says that every one of them is a combination of the elementary ones — the sum, the sum of pairs, the sum of triples, the product. So the list of coefficients does not merely contain those; it contains, in principle, everything about the roots that can be said without naming any of them.

Anything that does distinguish the roots is out of reach. Which root is largest, for instance, is not a symmetric quantity, and no formula in the coefficients delivers it. The permutation group of the roots is what stands between the coefficients and the roots themselves, and the study of that group is what eventually settles which polynomials can be solved by radicals.

The average that cannot be moved #

The relation for the sum has a picture, and it is the one marked in the first figure.

The average of the roots is −b/n, where b is the coefficient of the second-highest power. So changing the constant term of a polynomial — pushing its graph up and down — moves every root, sometimes drastically, and cannot shift their centre of mass by a hair. The roots may scatter, may collide, may leave the real line in pairs, and their average stays put.

That is a genuinely useful piece of information about an object nobody has solved. It is also the reason for the standard first step in solving a cubic: substituting x → x − b/3 shifts the centre of mass to the origin and kills the squared term, leaving a depressed cubic with two coefficients instead of three. Every classical solution method starts there, and what the substitution is really doing is moving the roots so that their average is zero.

One number that counts the real roots #

The second half of what the coefficients know is a matter of signs rather than sums.

For a depressed cubic x³ + px + q, the quantity −4p³ − 27q² is called the discriminant. It is a symmetric function of the roots in disguise — it is the product of the squared differences between them — and its sign settles a question that looks as though it needs the roots to answer.

The reasoning behind it is short. Complex roots of a real polynomial come in conjugate pairs, so a real cubic has either three real roots or one real root and a conjugate pair. The product of the squared differences is a positive real number in the first case — every difference is real, so every square is positive — and in the second case one of the differences is imaginary, its square is negative, and the whole product changes sign.

Two ways of getting the same fact, and the figure runs both: it finds the roots of three hundred and twenty-one cubics numerically and counts how many are real, and it computes the discriminant from the two coefficients. Every sample is required to agree, and the boundary between the two regimes is required to be where the discriminant passes through zero.

At exactly zero the polynomial has a repeated root, which is the case where the two regimes meet: a pair of real roots colliding and about to leave the real line together.

What it is used for #

Three uses, each of which is a case of extracting an answer without solving.

Deciding a shape without solving. Whether a cubic curve crosses the axis once or three times is answered by a sign, which is a computation on two numbers rather than a root-finding problem. The same works for the discriminant of a quadratic, where b² − 4ac is the familiar case and the same argument applies.

Bounding the roots. Since the roots sum to −b and multiply to ±d, no root can be enormous unless another compensates. A whole family of bounds — the largest root is at most one plus the largest coefficient in size, and similar — comes from these relations, and they are what a numerical root-finder uses to know where to start looking.

Recognising rational roots. If a polynomial with whole-number coefficients has a rational root, the product relation forces its numerator to divide the constant term and its denominator the leading coefficient. That leaves a finite list to test, and testing it is how a rational root is found in practice — a search made finite by an identity rather than by cleverness.

The same two numbers, one field over #

The relations turn up somewhere they were not put, and the coincidence is worth following because it explains two quantities that are otherwise introduced by decree.

A square matrix has a characteristic polynomial, whose roots are its eigenvalues — the directions the matrix leaves pointing where they were, together with the factors it stretches them by. For a two-by-two matrix that polynomial is λ² − (a + d)λ + (ad − bc), and Vieta’s relations then say something immediately:

The eigenvalues add to a + d, the sum down the diagonal. They multiply to ad − bc, the determinant.

So the determinant is not merely a number that measures area; it is the product of the eigenvalues, and the trace — the diagonal sum, which otherwise looks like an arbitrary thing to compute — is their sum. Both are symmetric functions of a set of numbers that may be complex, may be irrational, and may be a great deal of work to find, and both are readable off the matrix by inspection.

The two halves fit together exactly. The winding argument establishes that n roots exist and where none of them are; the symmetric functions say what they total. Between them there is a complete description of a set of numbers that has not been computed, which is a stranger position to be in than it first appears.

What the relations cannot do #

The relations describe the roots collectively and say nothing about any of them individually, and that limit is not a gap waiting to be filled.

Knowing the sum and the product of two numbers determines them — they are the roots of a quadratic, which can be solved. Knowing the three elementary symmetric functions of three numbers also determines them, in the sense that they are the roots of a cubic, which is exactly the polynomial that was started from. So the relations are complete and circular: they say everything about the roots that is sayable without labelling them, and turning that back into the roots is the original problem.

For degree five and above the circularity becomes a wall. The roots are determined by the coefficients, they exist, and there is no formula in radicals that produces them. The relations still hold, the sum is still −b, and the individual roots are still out of reach of any finite expression built from the coefficients with arithmetic and root extraction.

The completed square is the point where the relations can still be inverted by hand. The sum of the two roots is −b, so their average is −b/2, and the two of them sit symmetrically either side of it at a distance the constant term fixes. That is the quadratic formula, derived from Vieta’s relations and a picture rather than by algebraic manipulation, and the reason it stops working at higher degree is that no such symmetric arrangement is available.

What it costs #

Computing the elementary symmetric functions from the roots costs almost nothing; going the other way is the whole difficulty of numerical algebra.

The figures here find roots by Durand–Kerner iteration, which starts every root at a different point of a circle and pushes each one away from the others while pulling it toward the curve. It converges to all n roots at once, it needs no derivatives, and it takes a few hundred iterations to reach twelve significant figures on the small cases drawn here. Every root produced is then substituted back into the polynomial and required to give something below a billionth in size, which is the only honest way to trust an iterative answer.

The classical alternative — deflate the polynomial by dividing out each root as it is found — is faster and is a known trap: the errors in the first root propagate into the coefficients of the deflated polynomial, so the later roots are found for a slightly different polynomial than the one asked about. Vieta’s relations provide the check that catches it. If the computed roots do not sum to −b and multiply to ±d, something has gone wrong, and the comparison costs n additions.

That is the practical value of a relation that adds no new information: it is a cross-check, computable from a different direction, and cross-checks are what makes a numerical answer worth anything.

What the picture cannot show #

The figures draw roots in the plane, which requires them to be at specific places, and every general statement here is about polynomials whose roots are unknown. A drawing can only ever illustrate the relations on a case where the roots have already been found.

The discriminant figure sweeps one coefficient of one family, and the theorem is about every real cubic. Sweeping is evidence about the family swept; the argument about conjugate pairs is what covers the rest.

And there is a dimensional dishonesty in every picture of a root. The roots of a real polynomial live in the plane and the polynomial itself is a function from the plane to the plane, which needs four dimensions to graph. What the figures show is the domain with some points marked on it, and the function that made them is nowhere in the picture.

The ladder from here #

Rungs above: Newton’s identities, which convert between the elementary symmetric functions and the power sums, and are how the sum of the squares of the roots is computed without finding them. The resultant of two polynomials, which decides whether they share a root by a determinant. The discriminant in general degree, and its interpretation as the resultant of a polynomial and its derivative. The symmetric group acting on the roots, which is Galois theory’s starting point and the reason degree five is different. Sturm’s theorem, which counts the real roots in an interval exactly, by a sequence of polynomial divisions. And root bounds in earnest — Cauchy’s, Fujiwara’s — which are what any serious root-finder starts from.

The shape of the idea #

The pattern here is worth naming because it appears whenever a hard object has an easy summary: extract from the object the quantities that are unchanged by the symmetries nobody has broken, and expect those quantities to be computable when the object is not.

The roots have a symmetry — they can be relabelled — and the coefficients are precisely the functions that survive it. The winding number is what survives deformation of a loop. A colouring count is what survives the moves on a knot diagram. In each case the invariant is cheap and the object is expensive, and the invariant is worth having exactly because it can be got at from the wrong end.

What is unusual about this instance is how complete it is. The symmetric functions do not merely describe some of what the roots hold in common; they describe all of it, and the fact that recovering the roots from them is impossible at degree five is not a failure of the summary but a discovery about the symmetry.