Find the Zeros Calculator

What the calculator gives you

Enter a polynomial in x, in whatever shape you have it: multiplied out, as a product of factors, or as an equation with terms on both sides. The calculator rewrites it in standard form and returns every zero, including the complex ones, with its multiplicity and a check that it really makes the polynomial equal zero.

Exact answers come first. Rational zeros are found with the Rational Zero Theorem and tested in whole-number fraction arithmetic, so a zero such as 1/2 is reported as 1/2 rather than 0.4999999. Zeros of a quadratic factor are written in radical form — √2, 2 ± 3i, (−1 + i√3)/2 — with the square root simplified. Only a factor that has no exact split at all is solved numerically, and those zeros carry a ≈ sign so nobody mistakes them for exact values.

Below the answer you get the zeros sorted into real, complex, purely imaginary and rational groups, the factored form with each factor set equal to zero, the full list of possible rational zeros with the test result for each, synthetic division for every rational zero found, a numbered step-by-step solution, and a graph with the real zeros marked.

Zero, root, x-intercept

A zero of a function f is an input x with f(x) = 0. The same number gets a different name depending on what you are looking at:

• Zero — a property of the function: "3 is a zero of f".
• Root — a solution of the equation f(x) = 0: "the equation x² − 9 = 0 has roots ±3".
• x-intercept — a point on the graph: "the curve crosses the x-axis at (3, 0)".

The first two apply to every zero, real or complex. The third only applies to real zeros, because a complex number has no position on the x-axis. That single difference explains why a graph can show a cubic crossing the axis once while the algebra insists there are three zeros.

Finding the zeros of a function

For any function the task is the same — solve f(x) = 0 — but the tools depend on the kind of function. Polynomials are the case with a complete toolkit. You always know how many zeros to look for (the degree tells you), there is a finite list of fractions that could be rational zeros, the quadratic formula finishes off any quadratic factor, and every answer can be substituted back to confirm it.

That is why this calculator handles polynomials only. A function such as sin x − x/2 or eˣ − 3 needs different reasoning, often purely numerical, and an input like that is refused with an explanation rather than answered with a guess. If your function is a polynomial that has been written with brackets, fractions or on both sides of an equals sign, it is still a polynomial, and it can be typed exactly as it appears in your worksheet.

Finding all zeros of a polynomial

The calculator works in the same order a careful student would, and the step-by-step solution follows that order line by line:

1. Standard form. Expand products, move everything to one side and collect like terms. The degree n is now visible, and so is the number of zeros to find.
2. Whole numbers. Multiply through to clear any fractions or decimals, and divide out any common factor. Scaling a polynomial never changes its zeros.
3. Factor out x. If the constant term is 0, x is a factor and 0 is a zero. Repeat while the constant term is still 0; the number of repeats is the multiplicity of 0.
4. Possible rational zeros. Apply the Rational Zero Theorem to list every ±p/q.
5. Test and divide. Substitute each candidate. Each one that gives 0 is divided out by synthetic division, repeatedly, until the remainder is no longer 0.
6. Solve what remains. A leftover quadratic goes to the quadratic formula. A longer leftover is checked for repeated factors and for hidden quadratic factors, and only what resists all of that is solved numerically.
7. Check. Every zero is substituted back into the original polynomial, and the factors are multiplied back together to make sure they reproduce it.

Rational zeros and the Rational Zero Theorem

The Rational Zero Theorem (also taught as the Rational Root Theorem) applies to a polynomial with whole-number coefficients. If p/q is a zero in lowest terms, then p divides the constant term and q divides the leading coefficient. Listing every fraction ±p/q built that way gives the possible rational zeros. The theorem promises that any rational zero is somewhere on that list — not that everything on the list is a zero.

If you are listing the possible rational zeros by hand, the fiddly part is finding every factor of the constant term and the leading coefficient. The prime factorization calculator breaks a number into primes, from which every factor follows, and the GCD calculator reduces a candidate like 6/4 to lowest terms so duplicates are easy to spot.

Synthetic division

Once a candidate c checks out, dividing by (x − c) lowers the degree by one. Synthetic division does it with the coefficients alone: bring the first coefficient down, multiply by c, add to the next coefficient, and repeat. The last number is the remainder, which is f(c) itself, so a remainder of 0 both confirms the zero and hands you the quotient. For the example above, dividing by x + 2 (c = −2):

The bottom row, 2, −7, 3, is the quotient 2x² − 7x + 3. When c is a fraction p/q, the quotient of a whole-number polynomial still comes out with whole-number coefficients, which is what lets the calculator search the smaller polynomial again — and dividing by the same c a second or third time is exactly how a repeated rational zero reveals its multiplicity.

Real, complex and imaginary zeros

A real zero is a real number: rational like −2 or 1/2, or irrational like √2. A complex zero has the form a + bi with b ≠ 0, and a purely imaginary zero is the special case a = 0, such as 2i. Many courses use "imaginary zeros" loosely for every non-real zero; the calculator keeps the two apart, listing all non-real zeros as complex and flagging the purely imaginary ones.

When every coefficient is real — always the case here — non-real zeros arrive in conjugate pairs. If 2 + 3i is a zero, so is 2 − 3i. That is why a cubic with real coefficients has either one real zero or three, never two.

To find imaginary zeros by hand, reduce the polynomial to a quadratic factor whose discriminant b² − 4ac is negative, then apply the quadratic formula with √−1 = i. For x³ − x² + 4x − 4, grouping gives x²(x − 1) + 4(x − 1) = (x − 1)(x² + 4). Setting x² + 4 = 0 gives x² = −4, so x = ±2i. The zeros are 1, 2i and −2i. For x² − 4x + 13 the discriminant is 16 − 52 = −36, and x = (4 ± 6i)/2 = 2 ± 3i.

Zeros and multiplicity

The multiplicity of a zero is the number of times its factor divides the polynomial. In (x − 2)³(x + 1), which expands to x⁴ − 5x³ + 6x² + 4x − 8, the zero 2 has multiplicity 3 and −1 has multiplicity 1. The degree is 4, and 3 + 1 = 4: multiplicities always add up to the degree.

Multiplicity is visible on the graph at real zeros. With an odd multiplicity the curve crosses the x-axis; at multiplicity 3 or more it flattens as it crosses. With an even multiplicity it touches the axis and turns back. So (x − 2)³(x + 1) crosses at −1 normally and crosses at 2 with a flat, S-shaped pass.

The calculator never infers multiplicity from two decimals that happen to be close. A rational zero's multiplicity is the number of synthetic divisions that leave remainder 0. For the rest, it computes the greatest common divisor of the polynomial and its derivative in exact arithmetic: a repeated factor divides both, so that gcd exposes it, and its power is the multiplicity. That is how (x² + 1)² is reported as ±i, each with multiplicity 2.

Zeros in factored form

Factored form is the easy case, because of the zero product property: a product is zero only when at least one of its factors is zero. So the zeros of a product are the zeros of its factors, and each factor can be solved on its own.

You can type a factored polynomial straight into the calculator. It is expanded internally, because the Rational Zero Theorem and the checks work on standard form, and then the factored form is rebuilt from the zeros it finds and shown with each factor set to zero — which also makes it a way to factor a polynomial you only have in expanded form.

When no rational zero exists

Having no rational zero is the normal case, not a failure. What happens next depends on what is left:

• A quadratic factor is solved exactly with the quadratic formula, with the square root simplified. For x³ − 3x² − 2x + 6, the candidate 3 works, synthetic division leaves x² − 2, and the zeros are 3, √2 and −√2.
• A longer factor is solved numerically first, and pairs of the numerical zeros are then used to guess quadratic factors with rational coefficients. A guess counts only if exact division leaves remainder 0. For example, x⁴ + 4 has no rational zero, but it splits exactly as (x² + 2x + 2)(x² − 2x + 2), giving the exact zeros 1 ± i and −1 ± i. Likewise x⁴ − 5x² + 6 becomes (x² − 2)(x² − 3) with zeros ±√2 and ±√3.
• A factor with no such split, like x³ − 2, keeps its numerical zeros: x ≈ 1.259921 (the cube root of 2) and x ≈ −0.629961 ± 1.091124i. These come from the Durand–Kerner method, which improves estimates of all the zeros at once, polished with Newton's method. Sturm's theorem, computed exactly, fixes how many of them are real, so a real zero can't be misreported as complex because of a tiny rounding error in its imaginary part.

How to check a zero

The test for a zero is the definition: substitute it and see whether f gives 0. The calculator does this for every answer and says how in the Check column. A rational zero is substituted in exact fraction arithmetic, so the result is exactly 0. A zero from a quadratic factor is confirmed by dividing f(x) by that factor and getting remainder 0. A numerical zero cannot give exactly 0, so the size of f at that point is shown instead — typically around 10⁻¹⁵, which is rounding error.

The answer as a whole is checked two more ways, both shown in the step-by-step solution. Vieta's formulas say that for aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀, the n zeros counted with multiplicity add up to −aₙ₋₁/aₙ and multiply to (−1)ⁿ·a₀/aₙ. For 2x³ − 3x² − 11x + 6, the zeros −2, 1/2 and 3 add up to 3/2, which is −(−3)/2, and multiply to −3, which is −6/2. Descartes' Rule of Signs (below) is compared with the number of positive and negative real zeros actually found. A zero that was missed or counted twice would break at least one of these.

To check a zero yourself, substitute it into the original polynomial, not into a quotient from partway through the working; an arithmetic slip in an earlier step would otherwise go unnoticed. For a fraction zero, the fraction calculator is a quick way to do that arithmetic exactly.

Zeros on a graphing calculator

On a graph, the real zeros are the x-intercepts, so a graphing calculator finds them by locating where the curve meets the x-axis. On the TI-84 family the sequence is: enter the function under Y=, press GRAPH, then 2nd, CALC and choose 2:zero. The calculator asks for a left bound and a right bound on either side of one x-intercept, then a guess, and reports that zero. Repeat for each intercept. Other models have an equivalent root or zero command.

Two limits are worth knowing. A graph only ever shows real zeros — the conjugate pair of a cubic like x³ − 2 is invisible on it — and it reports decimals, so 1/2 appears as 0.5 and √2 as 1.4142136. A graph is still a useful companion to the algebra: an intercept at 0.5 tells you which of the possible rational zeros to test first. This page draws the graph of your polynomial under the result, with every real zero marked.

Degree, Descartes and a quick sanity check

By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n zeros once complex zeros are allowed and each is counted with its multiplicity. The stats row under the result adds real and complex zeros back together so that total can be compared with the degree directly.

Descartes' Rule of Signs gives a second check. Count the sign changes between consecutive nonzero coefficients of f(x): the number of positive real zeros is that count, or less than it by an even number. Doing the same for f(−x) bounds the negative real zeros. For x³ − 6x² + 11x − 6 the signs go +, −, +, −, so there are 3 or 1 positive real zeros; f(−x) has no sign changes, so there are no negative ones. The actual zeros are 1, 2 and 3.

Common mistakes

Treating the candidate list as the answer

The possible rational zeros only say where a rational zero could be. Each one still has to be tested, and usually most of them fail.

Stopping after the rational zeros

If the rational zeros account for fewer zeros than the degree, the rest are irrational or complex. Divide the rational ones out and solve the quotient; the quadratic formula often finishes the job.

Losing a repeated zero

After dividing out a zero, test it again on the quotient. In (x − 2)³ the zero 2 appears once in the list of distinct zeros but accounts for three of the polynomial's zeros.

Reading a decimal as exact

1.414214 is not √2; it is a rounded value of it. The calculator writes exact forms wherever they exist and puts ≈ before every numerical value so the difference stays visible.

Limitations

Only polynomials in the single variable x are accepted, up to degree 10, with each typed number at most 1,000,000,000 and, after expanding and clearing fractions, each coefficient at most 10¹². Zeros are written exactly when they are rational or come from a quadratic factor with rational coefficients. The general cubic (Cardano) and quartic (Ferrari) formulas are not used: their output is often unwieldy nested radicals, and for a cubic with three irrational real zeros it requires complex cube roots even though every answer is real. So a factor of degree three or more with no rational or quadratic split, such as x³ − 2 or x³ − 3x + 1, gets numerical zeros to six decimal places, marked ≈, even where a radical formula exists. When the list of possible rational zeros would be longer than 20,000, it is not tested one by one; rational zeros are then found from the numerical zeros and confirmed exactly, and the page says so.

Sources and further reading

• Zeros of Polynomial Functions — OpenStax College Algebra 2e (Rice University): the Rational Zero Theorem, the Factor Theorem, the Fundamental Theorem of Algebra and Descartes' Rule of Signs, with exercises.
• Zeroes/Roots of Polynomials and Finding Zeroes of Polynomials — Paul's Online Math Notes, Lamar University: multiplicity, and the rational-root process with synthetic division.
• Finding the Zero of a Function on the TI-84 Plus CE — Texas Instruments' own instructions for the graphing-calculator method above.
• Durand–Kerner method — general reference for the numerical method used on factors with no exact split.

Frequently asked questions