How many roots does a polynomial have?

Exactly as many as its degree, counted with multiplicity, provided complex roots are included. A cubic has three, a quartic four. This is the fundamental theorem of algebra, and the complex part matters: x² + 1 has degree two and no real roots at all, but two complex ones. A repeated root counts as many times as it repeats, so (x−1)³ has three roots that all happen to sit at the same place.

Why does the quadratic formula sometimes give a wrong answer?

Because of cancellation. When b² is much larger than 4ac, the square root is very close to |b|, so one of the two roots is found by subtracting two nearly identical numbers — and almost every significant figure disappears. On x² − 200000x + 1 the textbook form is wrong in the sixth digit; with a coefficient of 10⁹ it returns zero. The arithmetic is not at fault, the arrangement of it is.

What is the stable way to solve a quadratic?

Compute whichever root involves addition rather than subtraction, then get the other from the fact that the roots multiply to c/a. Take q = −(b + sign(b)√(b²−4ac))/2; the sign choice guarantees the two terms add rather than cancel. Then one root is q/a and the other is c/q. It is the same formula rearranged, it costs nothing extra, and it is exact on cases where the textbook version loses everything.

Is there a formula for cubics and quartics?

Yes for both, though neither is pleasant. Cardano published the cubic method in 1545 and Ferrari the quartic shortly after. Both are exact and both are numerically awkward — the cubic formula famously routes through complex numbers even when all three roots are real, which is how complex numbers first became unavoidable. In practice numerical methods are used for both, because they are shorter, more stable and extend to any degree.

Why is there no formula for degree five?

Because none can exist. Abel proved in 1824 that no expression built from the coefficients using arithmetic and roots can solve the general quintic, and Galois soon explained the underlying reason — it comes down to the structure of the symmetry group of the roots, which changes character at degree five. Particular quintics are still solvable exactly; x⁵ − 32 has the obvious root 2. It is the general case that is impossible.

Why do repeated roots come out imprecise?

Because the curve is flat where it touches, so a wide range of x values all produce something indistinguishable from zero. The relationship is sharp: a root of multiplicity m can only be located to about the machine precision raised to the power 1/m. A double root reaches around 10⁻⁸ and a triple root around 10⁻⁵, against 10⁻¹⁶ for a simple root. That is a property of the arithmetic rather than of any particular algorithm — no method in double precision does better.

Why do complex roots always come in pairs?

Because the coefficients are real. Taking the complex conjugate of the whole equation leaves the coefficients unchanged, so if a value satisfies the polynomial then so does its conjugate — same real part, opposite imaginary part. One consequence is that a polynomial of odd degree with real coefficients always has at least one real root, since an odd number of roots cannot be divided into pairs without one left over. That is why every cubic crosses the axis somewhere.

How sensitive are roots to the coefficients?

Far more than people expect. The standard demonstration is Wilkinson’s polynomial — (x−1)(x−2)…(x−10) expanded out. Change the coefficient of x⁹ from −55 to −55.0000001, a relative change of about two parts in a billion, and the roots shift by roughly a hundred thousand times that in relative terms. The arithmetic is fine; the problem is that coefficient form encodes the roots very indirectly, so a small change to the encoding is a large change to what it encodes.

What does multiplicity look like on the graph?

An odd multiplicity crosses the axis and an even one touches it and turns back. A simple root passes straight through; a double root sits at a minimum or maximum exactly on the axis; a triple root crosses but flattens out as it does so, with an inflection at the crossing. So a curve that comes down to zero and returns the way it came has a root of even multiplicity there, which is often quicker to see than to compute.

What does the residual tell me?

How far the polynomial actually is from zero at the value reported. Around 10⁻¹⁶ relative to the coefficients means exact to the limit of the arithmetic. A larger figure is not automatically an error — a repeated root cannot get that close, and a polynomial with very large coefficients will have proportionally larger residuals. What it does catch is a root that has not converged, which would show as a residual many orders of magnitude above its neighbours.

Polynomial Root Calculator

Every root of a polynomial, real and complex, with each one substituted back so you can see how well it actually satisfies the equation. A polynomial of degree n has exactly n roots counted with multiplicity — that is the fundamental theorem of algebra, and it only holds once complex numbers are al...

Every root, real and complex, with the working checked. A polynomial of degree n has exactly n roots counted with multiplicity — that is the fundamental theorem of algebra, and it holds only once complex numbers are allowed. Each root here is substituted back so you can see how well it actually satisfies the equation.

THE POLYNOMIAL

DEGREE

x³+x²+x+= 0

3 DISTINCT ROOTS

Found by Durand–Kerner iteration. The worst residual is 8.88e-16 — that is how far the polynomial is from zero at these values.

Root	Kind	Multiplicity	\|p(root)\|
1	real	1	4.8e-35
2	real	1	8.9e-16
3	real	1	2.0e-31

Roots are far more sensitive to the coefficients than people expect. The standard demonstration is Wilkinson’s polynomial, (x−1)(x−2)…(x−10), expanded out. Change the coefficient of x⁹ from −55 to −55.0000001 — a relative change of about two parts in a billion — and the roots move by around a hundred thousand times that in relative terms. Nothing is wrong with the arithmetic; the problem is that a polynomial written in coefficient form encodes its roots very indirectly, and small changes to the encoding make large changes to what it encodes.

WHERE THE ROOTS ARE

A real root is where the curve meets the axis. A root of even multiplicity touches the axis and turns back without crossing; an odd one passes through. That is a useful thing to know by eye — a curve that grazes zero and returns has a repeated root there.

Durand–Kerner iteration · live calculation

Created with❤️byeaglecalculator.com

HOW TO USE

1
Pick the degree, then enter every coefficient from the highest power down. A missing term is a zero and must be written in — leaving a gap changes the degree.
2
Read the multiplicities as well as the values. A root listed with a multiplicity of two or more is where the curve touches the axis and turns back rather than crossing it.
3
Check the residual column. It shows how far the polynomial is from zero at each root — around 10⁻¹⁶ means exact to the limit of the arithmetic, and a repeated root will legitimately be larger.
4
Treat any warning about unresolved roots as real. It means several values sit closer together than double precision can separate, and no method could do better with the same inputs.

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Last updated: August 3, 2026 · Stable closed forms to degree 2, Durand–Kerner above · A repeated root of multiplicity m resolves only to about ε^(1/m).