About the Polynomial Roots Calculator
This polynomial roots calculator finds all the zeros of a polynomial — real and complex — up to degree 12. Type the polynomial in ordinary notation (for example 2x^3 − 3x^2 − 11x + 6 or (x − 1)^2(x + 2)) and you get every root with its multiplicity, the rational roots in exact fraction form, the complex conjugate pairs, the factored form over the rationals and a graph of the polynomial showing where it crosses the x-axis.
It is meant for algebra and precalculus students working through the rational root theorem, for checking factoring homework, and for anyone who needs the roots of a characteristic polynomial or a higher-degree equation that has no simple formula. The rational root test candidates ±p/q are listed so you can see which ones actually work.
The method combines exact and numeric algebra: repeated factors are separated exactly (square-free factorisation), rational roots are confirmed with exact fraction arithmetic, and the remaining roots are found numerically with the Durand–Kerner method and polished with Newton’s method, so irrational and complex roots are accurate to about 6 decimal places.
How to use the polynomial roots calculator
- 1Type the polynomial, e.g. 2x^3 - 3x^2 - 11x + 6.
- 2Read the list of all roots; repeated roots show their multiplicity.
- 3Check the exact rational roots and the factored form.
- 4Compare with the rational root test candidates to see which ±p/q work.
- 5Use the graph to see where the polynomial crosses or touches the x-axis.
Formula and method
By the fundamental theorem of algebra a polynomial of degree n has exactly n complex roots counted with multiplicity, and non-real roots of a real polynomial come in conjugate pairs a ± bi. The rational root theorem says any rational root p/q in lowest terms of an integer-coefficient polynomial has p dividing the constant term a₀ and q dividing the leading coefficient aₙ, which gives a finite list of candidates.
This calculator first splits P into square-free factors (using gcd(P, P′)) so repeated roots are detected exactly and their multiplicity is known. Each factor is searched for rational roots, each verified with exact fraction arithmetic and divided out. What remains is solved numerically with the Durand–Kerner simultaneous iteration and refined with Newton’s method.
- aₙ, a₀
- Leading coefficient and constant term
- rᵢ
- Roots (zeros) of P
- p/q
- Rational root candidate in lowest terms
Worked examples
x³ − 6x² + 11x − 6
The candidates are ±1, ±2, ±3, ±6. Testing gives P(1) = 1 − 6 + 11 − 6 = 0, P(2) = 8 − 24 + 22 − 6 = 0 and P(3) = 27 − 54 + 33 − 6 = 0, so the cubic factors completely as (x − 1)(x − 2)(x − 3).
2x³ − 3x² − 11x + 6 with a fractional root
With a₀ = 6 and aₙ = 2 the candidates are ±1, ±2, ±3, ±6, ±1/2 and ±3/2. Three of them work: −2, 1/2 and 3, giving 2x³ − 3x² − 11x + 6 = (x + 2)(2x − 1)(x − 3).
x⁴ − 1 has two real and two complex roots
x⁴ − 1 = (x² − 1)(x² + 1) = (x − 1)(x + 1)(x² + 1). The real roots are ±1, and x² + 1 = 0 gives the conjugate pair ±i.
Repeated root: x³ − 3x + 2
x³ − 3x + 2 = (x − 1)²(x + 2). Numerical methods alone would return two slightly different values near 1; the exact square-free step shows it is a double root, where the graph touches the axis.
Irrational and complex roots: x³ − 2
The only real root is the cube root of 2 ≈ 1.259921, which is irrational (none of ±1, ±2 work). The other two roots are 2^(1/3)·(−1/2 ± (√3/2)i) ≈ −0.629961 ± 1.091124i.
Frequently asked questions
How many roots does a polynomial have?+
A polynomial of degree n has exactly n roots in the complex numbers when each is counted with its multiplicity (the fundamental theorem of algebra). Some or all of them may be non-real, and those come in conjugate pairs.
What is the rational root theorem?+
If a polynomial has integer coefficients, every rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient. Listing all ±p/q gives the only possible rational roots, which you then test.
What does multiplicity of a root mean?+
Multiplicity is how many times a factor (x − r) appears. For (x − 1)²(x + 2), x = 1 has multiplicity 2. At an even-multiplicity root the graph touches the x-axis and turns back; at an odd one it crosses.
Can a polynomial have no real roots?+
Yes, but only if its degree is even, e.g. x² + 1 or x⁴ + x² + 1. Every odd-degree real polynomial has at least one real root because its graph goes from −∞ to +∞ and must cross the axis.
How accurate are the irrational and complex roots?+
Rational roots are exact. Irrational and complex roots are computed numerically and polished with Newton’s method, so they are shown to 6 decimal places and are typically accurate well beyond that.