Algebraic Multiplicity Calculator
Two polynomials: x² − 4, and (x − 2)²(x + 2)². Ask either where it equals zero and the answer is the same — at 2 and at −2. Yet the graphs look nothing alike. The first slices straight through the axis at both points. The second comes down, touches, and turns back the way it came.
The zeros are identical. What separates the pictures is multiplicity — how many times each zero repeats inside the factored form. The algebraic multiplicity calculator above finds every zero, states its multiplicity, and draws the curve so you can see the difference rather than take it on trust.
What Is the Multiplicity of a Zero?
Write a polynomial as a product of factors. The multiplicity of a zero is the exponent on its factor — how many times that factor appears.
f(x) = (x + 3)(x − 2)²(x + 1)³
Setting each factor to zero gives x = −3, x = 2 and x = −1, with multiplicities 1, 2 and 3 — the exponents above them. When a polynomial arrives factored, that is the whole job.
A zero of multiplicity 1 is simple. Multiplicity 2 or more makes it repeated: "double zero" means 2, "triple" means 3. The word algebraic matters in one specific context, covered further down.
Multiplicity and the Graph: Cross or Touch?
This is what multiplicity is for, and the rule has no exceptions:
Odd multiplicity → the graph crosses the x-axis.
Even multiplicity → the graph touches and turns back.
The reason is worth ten seconds. Near a zero at x = r, the polynomial behaves like (x − r) raised to the multiplicity. That bracket is negative just left of r and positive just right of it. An odd power keeps a negative number negative, so the sign flips as you pass through — and a sign flip is exactly what crossing means. An even power cancels the minus signs in pairs: same sign on both sides, no crossing possible, so the curve must bounce.
Apply it to the opening pair. In x² − 4 = (x − 2)(x + 2) both multiplicities are 1, odd, so it crosses twice. In (x − 2)²(x + 2)² both are 2, even, so it touches twice.
There is a second effect. The higher the multiplicity, the flatter the curve lies against the axis. Multiplicity 1 cuts through at a healthy angle; multiplicity 3 still crosses but levels out first and pushes through almost horizontally — the shape of y = x³ at the origin. That is why the sketch above separates a plain crossing from a flattened one.
Even and Odd Multiplicity at a Glance
| Multiplicity | At the axis | Shape to picture |
|---|---|---|
| 1 | crosses | a straight cut through, like a line |
| 2 | touches, turns back | a parabola tip resting on the axis |
| 3 | crosses, flattening | the S-bend of y = x³ at the origin |
| 4 | touches, flatter still | a broad, lazy bounce |
| 5 and up | follows the odd/even rule | flatter with every step up |
Only the parity decides between crossing and touching — never the size. Multiplicity 12 touches for the same reason 2 does; multiplicity 13 crosses for the same reason 1 does. Size controls only the flatness.
How to Find Multiplicity from a Factored Polynomial
Here the job is reading, not calculating. Three habits keep it error-free.
Set each factor to zero. (x + 3) gives x = −3, not 3 — the sign flips. (2x − 1) gives x = ½. This sign slip is the most common mistake in the topic.
The exponent is the multiplicity. No visible exponent means 1. In (x + 3)(x − 2)²(x + 1)³: −3 has multiplicity 1, 2 has multiplicity 2, −1 has multiplicity 3.
A bare x is a zero too. The factor x means x = 0 is a zero; x³ gives it multiplicity 3. Easy to skip because it carries no bracket.
One trap: if the same factor appears twice in different places — (x − 4)²(x − 1)(x − 4)⁴ — the exponents add. That zero has multiplicity 6, not 2 and not 4.
How to Find Multiplicity When the Polynomial Is Not Factored
An expanded polynomial hides its multiplicities. x³ − 6x² + 12x − 8 looks like an ordinary cubic; only factoring reveals (x − 2)³. By hand:
1. Find a zero. The rational root theorem gives the shortlist — the rational zeros calculator builds it for you.
2. Divide it out with synthetic division.
3. Test the same zero again. This is the step people skip, and it is where multiplicity lives. Keep dividing by the same zero until it stops working — the number of successful divisions is the multiplicity. For x³ − 6x² + 12x − 8, the zero 2 divides in three times.
4. Carry on with what is left, until the remaining factor is a quadratic the formula can finish.
The calculator runs this in exact whole-number fraction arithmetic. A tool testing zeros in decimals has to judge whether a remainder of 0.0000000001 counts as zero — and at a repeated zero that judgement is exactly what fails.
How to Use This Algebraic Multiplicity Calculator
Type it in whichever form you have. Factored input needs no expanding — (x+3)(x-2)^2(x+1)^3 goes straight in. So does the expanded version, in any term order, with fractions like (1/2)x^2 - 2x + 2 or decimals. Powers accept a caret, **, or pasted superscripts such as x².
Use the keypad, or your own keyboard. Phone keyboards bury the caret and brackets, so the tool brings keys for x, xⁿ, brackets and a fraction bar. The keyboard icon in the box switches your system keyboard on and off, and the line underneath echoes your input in textbook form.
Press = and read the top row. Each zero appears with a ×2 or ×3 badge when it repeats, followed by the polynomial as understood and the degree.
Then look at the sketch. The curve carries a dot on every real zero: blue where it crosses, orange where it touches and turns back. The picture and the verdict come from the same working, so they cannot disagree.
Open "Each zero in detail" for the multiplicity and behaviour of each zero, plus a count of every zero the polynomial has — real and complex — grouped by multiplicity. That count stays exact even for zeros the calculator cannot write down.
The step panel clears any fractions, shows the polynomial factored with the exponents visible, turns each exponent into graph behaviour, and finishes with the end behaviour of the arms.
Multiplicities Add Up to the Degree
A degree-n polynomial has exactly n zeros, counting complex ones and counting each zero as many times as its multiplicity. That is the fundamental theorem of algebra, and it is the best self-check in this topic.
For (x + 3)(x − 2)²(x + 1)³, the multiplicities 1, 2 and 3 add to 6 — the degree. If yours fall short, a zero or an exponent has been missed.
It also tells you when to stop. x³ + 2x² − 4x − 8 is (x + 2)²(x − 2): multiplicity 2 at −2 and 1 at 2, total 3, equal to the degree. Nothing is hiding.
Complex zeros count in the same total. x⁵ − x has five: 0, 1 and −1 on the real line, plus the pair i and −i — all simple, degree 5.
Algebraic Multiplicity vs Geometric Multiplicity
The word algebraic separates this from a different quantity of the same surname, and the distinction lives in linear algebra.
Eigenvalues come from a matrix's characteristic polynomial. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of that polynomial — exactly what this page describes. The geometric multiplicity is the number of independent eigenvectors for that eigenvalue, the dimension of its eigenspace.
They are not always equal. Geometric multiplicity is at least 1 and never exceeds the algebraic one, and the gap decides whether a matrix can be diagonalised: algebraic 2 with geometric 1 is what forces Jordan form.
Practically: expand the characteristic polynomial, type it in, and every multiplicity reported is the algebraic one. Geometric multiplicity cannot be computed from the polynomial at all — it needs the matrix, specifically the null space of A − λI — so no polynomial-only tool can give it, and this one does not pretend otherwise.
What This Calculator Does Differently
It draws the answer. Other tools hand you a list: "x = 2, multiplicity 2". Graphing sites do the reverse, drawing a curve and leaving the multiplicities to you. Here both sit together, from the same working, each zero colour-marked as a crossing or a bounce.
Factored input goes in as it stands, and the factored form is rebuilt in the working so the multiplicities appear as exponents, in the notation the question used.
Every zero is tested exactly. The arithmetic is whole-number fractions throughout, so 1⁄3 is tested as one third, never 0.333. There is no tolerance at which a remainder is called "close enough" — the decision that goes wrong at repeated zeros.
Awkward zeros are named. (x − 1)²(x² − 2) returns 1 with multiplicity 2 alongside the exact surds ±√2. Complex zeros are named too and kept off the sketch: (2x − 1)²(x² + 4) gives ½ with multiplicity 2 and the pair ±2i.
The count is exact even when the names are not. A square-free decomposition works out how many distinct zeros carry each multiplicity without finding them, so (x² + 1)³ is correctly reported as two zeros of multiplicity 3, and a polynomial with no closed-form zeros still gets an exact census.
Fractions are cleared, not refused. (1/2)x² − 2x + 2 is multiplied through — scaling moves no zeros — and returns 2 with multiplicity 2.
Worked Examples
| Polynomial | Zeros (multiplicity) | At the axis |
|---|---|---|
| (x + 3)(x − 2)²(x + 1)³ | −3 (1), −1 (3), 2 (2) | crosses, crosses flattening, touches |
| x² − 4 | −2 (1), 2 (1) | crosses at both |
| (x − 2)²(x + 2)² | −2 (2), 2 (2) | touches at both — same zeros, opposite graph |
| x³ − 6x² + 12x − 8 | 2 (3) | crosses, flattening — hidden until factored |
| x⁴ − 8x³ + 24x² − 32x + 16 | 2 (4) | touches, very flat |
| (x − 1)²(x² − 2) | 1 (2), ±√2 (1 each) | touches at 1, crosses at both surds |
| (2x − 1)²(x² + 4) | ½ (2), ±2i (1 each) | touches at ½; the complex pair is off the graph |
| (x² + 1)³ | ±i (3 each) | never meets the axis at all |
The third row is the one to keep: the same pair of zeros, and a graph that crosses in one case and bounces in the other.
Frequently Asked Questions
What is the multiplicity of a zero?
How do you find the multiplicity of each zero?
Does the graph cross or touch at a zero?
Why does an even multiplicity make the graph bounce?
What does multiplicity 3 look like on a graph?
Do the multiplicities have to add up to the degree?
What is the difference between algebraic and geometric multiplicity?
What is a repeated or double zero?
Can complex zeros have multiplicity?
How do you find multiplicity from an expanded polynomial?
What if the zeros are irrational?
Is this algebraic multiplicity calculator free?
Multiplicity is the difference between knowing where a graph meets the axis and knowing what it does when it gets there. Read the exponents when the polynomial is factored, divide repeatedly when it is not, check that the multiplicities reach the degree, and let the odd-even rule turn those numbers into a shape. When the zeros themselves are the hard part, the rational zeros theorem calculator finds them and the Descartes' rule of signs calculator says how many to expect — with the whole free toolkit on the Monkza calculators page.