Algebraic Multiplicity Calculator
Two polynomials: x² − 4, and (x − 2)²(x + 2)². Both are zero at 2 and at −2 — nowhere else. Yet the first slices straight through the axis at both points, while the second kisses it and turns back.
What separates the pictures is multiplicity — how many times each zero repeats inside the factored form. This algebraic multiplicity calculator finds every zero, states its multiplicity, and draws the curve so you can see the difference.
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. Multiplicity 1 is a simple zero; 2 or more is a repeated zero, "double" and "triple" meaning 2 and 3. The word algebraic matters in one 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 the axis 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 bounces.
Apply it to the opening pair. In x² − 4 both multiplicities are 1, odd, so it crosses twice. In (x − 2)²(x + 2)² both are 2, even, so it touches twice. Same zeros, opposite pictures.
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 — the shape of y = x³ at the origin.
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 |
Only the parity decides between crossing and touching — never the size. Multiplicity 12 touches for the same reason 2 does; 13 crosses for the same reason 1 does.
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 slip is the commonest mistake in the topic.
The exponent is the multiplicity. No visible exponent means 1. A bare x is a zero too: it means x = 0, and x³ gives that zero multiplicity 3.
Collect like factors first. If the same factor appears twice — (x − 4)²(x − 1)(x − 4)⁴ — the exponents add, so 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, and the rational zeros calculator builds it for you. 2. Divide it out by synthetic division. 3. Test the same zero again — this is the step people skip, and it is where multiplicity lives. Keep dividing by that zero until it stops working; the number of successful divisions is the multiplicity. 4. Carry on until the leftover 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 or decimals. The keypad carries x, xⁿ, brackets and a fraction bar, with a one-tap switch to your own keyboard.
Press = and read the band. Each zero appears with a ×2 or ×3 badge when it repeats, then the degree and how many distinct zeros were found. Below it the sketch: the curve with a dot on every real zero — blue where it crosses, orange where it touches and turns back. Picture and words come from the same working, so they cannot disagree.
Open "Each zero in detail" for every zero's multiplicity and behaviour, plus an exact count of all the zeros grouped by multiplicity — exact even for zeros the calculator cannot name. The step panel shows the polynomial factored with the exponents visible.
Multiplicities Add Up to the Degree
A degree-n polynomial has exactly n zeros, counting complex ones and counting each as many times as its multiplicity. That is the fundamental theorem of algebra, and 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), total 3, equal to the degree, so nothing is hiding. Complex zeros count in the same total: x⁵ − x has 0, 1, −1 and the pair ±i.
Algebraic Multiplicity vs Geometric Multiplicity
The word algebraic separates this from a different quantity of the same surname, which 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 it. They are not always equal: geometric is at least 1 and never exceeds algebraic, and a gap between them is what forces Jordan form instead of diagonalisation.
Practically: expand the characteristic polynomial, type it in, and every multiplicity reported is the algebraic one. Geometric multiplicity needs the matrix — 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 come from the same working, each zero colour-marked as a crossing or a bounce.
Every zero is tested exactly. The arithmetic runs on whole-number fractions, 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 a repeated zero.
It names the awkward zeros, and counts the rest. Irrational zeros arrive as exact surds: (x − 1)²(x² − 2) gives 1 with multiplicity 2 alongside ±√2. Complex zeros are named and kept off the sketch. And 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.
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 |
| (x − 2)²(x + 2)² | −2 (2), 2 (2) | touches at both — same zeros as x² − 4, opposite graph |
| x³ − 6x² + 12x − 8 | 2 (3) | crosses, flattening — hidden until factored |
| (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 |
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?
Do the multiplicities have to add up to the degree?
What is the difference between algebraic and geometric multiplicity?
Can complex zeros have multiplicity?
Is this algebraic multiplicity calculator free?
Multiplicity is the difference between knowing where a graph meets the axis and knowing what it does 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 end behavior calculator shows where the arms go.
