Polynomial Equations

You have solved quadratic equations. A quadratic has degree 2, and it always has exactly 2 roots — sometimes both real, sometimes a repeated pair, sometimes a complex conjugate pair. But always two.

Now take a cubic: x^3 - 6x^2 + 11x - 6 = 0. Try x = 1: 1 - 6 + 11 - 6 = 0. Try x = 2: 8 - 24 + 22 - 6 = 0. Try x = 3: 27 - 54 + 33 - 6 = 0. Three roots — 1, 2, 3 — for a degree-3 equation.

Is this a coincidence, or is there a pattern?

The pattern is one of the deepest facts in all of mathematics: a polynomial equation of degree n always has exactly n roots, once you count them the right way. "The right way" means allowing complex numbers and counting repeated roots according to their multiplicity. This fact is called the Fundamental Theorem of Algebra, and this article is about what it says, why you should believe it, and what it lets you do.

What a polynomial equation is

A polynomial equation is an equation of the form

where a_0, a_1, \ldots, a_n are constants (real or complex), and a_n \neq 0. The number n is the degree of the equation. You have already met the special cases:

The last column is the claim. Every row says: degree n, exactly n roots. The first two rows you can verify from formulas you already know. The general claim is the Fundamental Theorem.

The Fundamental Theorem of Algebra

The theorem can be stated in one sentence.

"Counted with multiplicity" means: if (x - r)^k divides the polynomial but (x - r)^{k+1} does not, then r is counted k times. The equation (x - 2)^3 = 0 has degree 3 and one distinct root (x = 2), but that root has multiplicity 3, so the count is 3.

The theorem says something remarkable: no matter how high the degree, no matter how complicated the coefficients, the polynomial always has enough roots. It cannot "run out" of solutions the way x^2 + 1 = 0 runs out of real solutions. Over the complex numbers, every polynomial of degree n factors completely:

where r_1, r_2, \ldots, r_n are the n roots (some possibly equal, some possibly complex).

The proof of this theorem is not elementary — it requires tools from complex analysis or topology that go well beyond school algebra. But the statement is something you can use right now, and it shapes every calculation in this article.

A consequence for real polynomials

If all the coefficients a_0, a_1, \ldots, a_n are real numbers, the complex roots come in conjugate pairs. That is, if \alpha + \beta i is a root (with \beta \neq 0), then \alpha - \beta i is also a root. This means a real polynomial of odd degree always has at least one real root — because the complex roots pair off, and an odd number of roots cannot all be paired.

Relation between roots and coefficients

Here is where the theorem becomes a working tool. Suppose the polynomial equation

has roots r_1, r_2, \ldots, r_n. By the Fundamental Theorem, the polynomial factors as

Expand the right side and compare coefficients. What falls out is a remarkable set of identities.

The pattern for any degree

Divide both sides by a_n to get the monic form (leading coefficient 1):

Expanding (x - r_1)(x - r_2) \cdots (x - r_n) is a systematic process. The coefficient of x^{n-1} comes from choosing "x" from all but one factor and "-r_i" from the remaining one. The coefficient of x^{n-2} comes from choosing "-r_i" from two factors and "x" from the rest. And so on.

The result, comparing the left side with the expanded right side:

The signs alternate: the sum of roots has a - sign, the sum of pairwise products has a +, the sum of triple products has a -, and so on.

Checking against what you know

For a quadratic ax^2 + bx + c = 0 with roots \alpha and \beta:

• Sum: \alpha + \beta = -b/a
• Product: \alpha \beta = c/a

These are exactly the formulas you already know from quadratic equations. The general relations are the same idea, extended to higher degrees.

For a cubic ax^3 + bx^2 + cx + d = 0 with roots \alpha, \beta, \gamma:

• Sum: \alpha + \beta + \gamma = -b/a
• Sum of pairwise products: \alpha\beta + \alpha\gamma + \beta\gamma = c/a
• Product: \alpha\beta\gamma = -d/a

Why the sign pattern alternates

Here is the intuition. Start with two roots and expand:

The coefficient of x picks up a minus sign (because each -r_i contributes one negative). The constant term is a product of two negatives, so it is positive. Now add a third root:

Each time you pick one more (-r_i) from the factors, you introduce one more minus sign. Picking one r_i gives (-1)^1, picking two gives (-1)^2 = +1, picking three gives (-1)^3 = -1. The signs alternate because each additional root chosen flips the sign once.

Two worked examples

Common confusions

• "A degree-n polynomial always crosses the x-axis n times." Not necessarily — it has n roots, but some may be complex (not on the real axis) and some may be repeated (the curve touches the axis without crossing). The equation x^2 + 1 = 0 has degree 2 but the graph y = x^2 + 1 never touches the x-axis; both roots are complex (\pm i).

• "The Fundamental Theorem says the roots are real." It says nothing about real roots. It guarantees n roots in the complex numbers. For real polynomials, some or all may happen to be real, but the theorem does not promise this.

• "Sum of roots = b/a." The sign is easy to forget. The sum of roots is -b/a (or more generally -a_{n-1}/a_n). Missing the minus sign is one of the most common errors in root-coefficient problems.

• "A polynomial of degree n can have more than n roots." It cannot. If you find n + 1 distinct values of x that make a degree-n polynomial equal to zero, you have made an arithmetic error somewhere. The only polynomial with infinitely many roots is the zero polynomial (the one where every coefficient is 0).

• "Multiplicity is just a bookkeeping trick." Multiplicity has geometric meaning. A root of multiplicity 1 is where the curve crosses the axis cleanly. A root of multiplicity 2 is where the curve touches the axis and bounces back (like y = x^2 at the origin). A root of multiplicity 3 is where the curve crosses the axis but flattens out as it does (like y = x^3 at the origin). The multiplicity controls the shape.

Going deeper

Why "fundamental"?

The theorem is called "fundamental" not because its proof is basic — the proof is genuinely hard — but because it is the foundation on which the entire theory of polynomials rests. Without it, you could not be sure that factoring a polynomial into linear factors is always possible. Every factoring method you use — the Factor Theorem, synthetic division, grouping — implicitly relies on the Fundamental Theorem guaranteeing that the roots exist in the first place.

The theorem was first proved rigorously in the early 19th century. Multiple proofs exist, using tools from complex analysis, topology, or algebra. The simplest conceptual argument uses the fact that a polynomial p(z) of degree n behaves like z^n for large |z|, and the function z^n wraps around the origin n times as z travels in a large circle — so by a topological argument (the "winding number"), p(z) must hit zero at least once. Repeated application then gives all n roots.

Beyond degree four: no general formula

For degrees 1 through 4, there exist closed-form formulas that express the roots in terms of the coefficients using only +, -, \times, \div, and n-th roots (radicals). The quadratic formula is the degree-2 case. There is a cubic formula (degree 3, much messier) and a quartic formula (degree 4, very messy).

At degree 5, something remarkable happens. A result proved independently by Abel and Galois shows that no such formula exists for the general quintic equation. It is not that nobody has found one — it is proved impossible. The operations +, -, \times, \div, and radicals are simply not powerful enough to express the roots of every fifth-degree equation.

This does not mean degree-5 equations have no roots — the Fundamental Theorem guarantees they have exactly five. It means you need other tools to find them: numerical methods, special functions, or symmetry-based techniques. The roots exist; they just resist being written down in closed form.

Symmetric functions of roots

The root-coefficient relations are instances of elementary symmetric polynomials. Given variables r_1, \ldots, r_n:

• e_1 = r_1 + r_2 + \cdots + r_n (sum)
• e_2 = \sum_{i < j} r_i r_j (sum of pairwise products)
• e_k = \sum_{i_1 < i_2 < \cdots < i_k} r_{i_1} r_{i_2} \cdots r_{i_k}
• e_n = r_1 r_2 \cdots r_n (product)

Newton's identities connect these to the power sums p_k = r_1^k + r_2^k + \cdots + r_n^k. For instance, p_2 = e_1^2 - 2e_2. This means you can compute r_1^2 + r_2^2 + \cdots + r_n^2 purely from the coefficients, without knowing the individual roots — a technique that appears frequently in competitive mathematics.

Where this leads next

Polynomial equations are the trunk from which many algebraic branches grow.

• Sum and Product of Roots — a focused treatment of the root-coefficient relations for quadratics, with applications to symmetric functions.
• Remainder and Factor Theorems — the key tools for testing whether a given value is a root and for peeling off known factors.
• Polynomial Factorization — breaking a polynomial into irreducible pieces using the Factor Theorem, grouping, and identities.
• Polynomial Operations — addition, subtraction, multiplication, and division of polynomials — the arithmetic that underlies every calculation in this article.
• Polynomials — Introduction — the definitions and terminology for polynomials, if you want to revisit the basics.