Zeros and Multiplicity: Cross, Touch, or Flatten

The graph

Change a multiplicity

exponent on (x + 2) — the zero at x = −2

exponent on (x − 1) — the zero at x = 1

exponent on (x − 3) — the zero at x = 3

Four polynomials, read zero by zeropick one, then step through it

Why odd crosses and even touches

This is not a rule to memorize. It follows from one fact about exponents: a negative number raised to an even power turns positive, and raised to an odd power stays negative.

Start from the factor theorem: r is a zero of f exactly when (x − r) divides f. The multiplicity of r is how many copies of that factor appear in the complete factorization. So if r has multiplicity m, we can always write

peel off the repeated factor f(x) = (x − r)^m · g(x), where g collects every other factor
and g(r) ≠ 0 — if it were 0, another (x − r) would still be hiding inside g

That last line is the whole reason multiplicity is well defined: you keep pulling out copies of (x − r) until none is left, and m is how many you pulled.

The local model

Now zoom in near x = r. Polynomials are continuous, so if x is close to r then g(x) is close to g(r) — a fixed non-zero number. Call it C. Then for x near r,

f(x) ≈ C(x − r)^m, C = g(r) ≠ 0

Every question about what the graph does at a zero is now a question about the simplest possible power function. Switch the live graph to Local model to watch the dashed copy of C(x − r)^m lie down on top of the real curve. Write u = x − r, so u is negative just left of the zero and positive just right of it:

at u = −0.2	u¹	u²	u³	u⁴
left of the zero	−0.2	+0.04	−0.008	+0.0016
right of the zero (u = +0.2)	+0.2	+0.04	+0.008	+0.0016
did the sign change?	yes	no	yes	no

Odd m: u^m is negative on the left and positive on the right, so f changes sign and the graph must pass through the axis — it crosses. Even m: u^m is positive on both sides, so f keeps the sign of C the whole time. It reaches 0 exactly at r and then goes back the way it came — it touches. The sign of C decides which side of the axis that touch happens on: C > 0 gives a local minimum sitting on the axis, C < 0 a local maximum.

Those four pictures are the entire vocabulary. Everything a polynomial does at an x-intercept is one of them, possibly flipped upside down by a negative C.

Why higher multiplicity looks flat

Parity settles cross-versus-touch. The size of m settles how flat the graph looks getting there, and again it is only arithmetic on small numbers:

distance u from the zero	u¹	u²	u³	u⁴
0.5	0.5	0.25	0.125	0.0625
0.2	0.2	0.04	0.008	0.0016
0.1	0.1	0.01	0.001	0.0001

A tenth cubed is a thousandth. So near a multiplicity-3 zero the output is a hundred times smaller than near a multiplicity-1 zero, and a hundred times smaller is indistinguishable from zero at graph resolution. The curve appears glued to the axis, then peels away.

There is an exact statement behind the impression. Differentiate f = (x − r)^mg(x) with the product rule:

f′(x) = m(x − r)^m−1g(x) + (x − r)^mg′(x)
at x = r: if m ≥ 2 both terms contain a positive power of (x − r), so f′(r) = 0
if m = 1: f′(r) = g(r) = C ≠ 0, so the crossing has a genuine slant

Every zero of multiplicity 2 or more has a horizontal tangent, and multiplicity 1 never does. But be careful with the converse: a horizontal tangent is not a turning point. At an odd multiplicity of 3 or more the curve flattens, kisses the tangent line, and keeps going in the same direction. Case 3 in the stepper shows exactly this — a slope of zero at a point the graph sails straight through.

Counting: multiplicities add up to the degree

A degree-n polynomial has exactly n zeros when they are counted with multiplicity and complex zeros are allowed. That is the Fundamental Theorem of Algebra, and it is the accounting rule that makes these problems finite. Two consequences worth having at hand:

The real zeros can account for less than n. A factor with no real zeros, such as x² + 1, still eats 2 degrees but puts no point on the x-axis. So in f(x) = (x − 1)(x² + 1) the degree is 3 while the graph has a single intercept. Such a factor is always positive or always negative, so it never affects the sign chart either — here x² + 1 ≥ 1 > 0 for every real x, and the sign of f is decided entirely by (x − 1).

The number of crossings has the same parity as the degree. The crossings are the zeros of odd multiplicity, and adding up all the multiplicities to get n means the count of odd ones must be even or odd along with n. That in turn matches the end behaviour: even degree sends both ends the same way, so the graph must return to the side it started on, which takes an even number of crossings. Use it as a free check — if you count 3 crossings on a degree-4 problem, something is misread.

Turning points are capped at n − 1. A polynomial of degree n changes direction at most n − 1 times, and each touch point is automatically one of them.

Reading a graph backwards, honestly

Given a picture, you can read the location of each zero and the parity of its multiplicity — and that is all. A touch could be multiplicity 2, 4, or 20; a flat crossing could be 3, 5, or 9. Flatness narrows it, and a stated degree usually pins it down, as in case 4 of the stepper. Without a stated degree, the correct answer names the smallest multiplicity consistent with the picture and says so.

What must be checked

Factor completely first. Multiplicity is only readable from a fully factored form. Consider f(x) = (x² − 4)(x − 2). It looks like three simple zeros, but the first factor is a difference of squares:

(x² − 4)(x − 2) = (x − 2)(x + 2)(x − 2) = (x − 2)²(x + 2)

So x = 2 has multiplicity 2 and the graph touches there rather than crossing. Anyone who read the unfactored version drew the wrong picture.

Domain: all real numbers. A polynomial has no variable in a denominator and none under a radical, so nothing is excluded and no value of x is illegal. There are no restrictions to state and no extraneous solutions manufactured by the factoring itself — the zero-product property is an equivalence, not a one-way implication.

Where extraneous and lost solutions really come from. If the polynomial arrived after clearing a denominator, any root that made that original denominator zero was never legal, however clean it looks — substitute every root back into the original equation. If it arrived after squaring both sides, squaring can turn a false statement true, so again check each root in the original. And the opposite error loses answers: never divide both sides by a factor containing x.

dividing by x destroys a solution x³ = 4x → divide by x: x² = 4 → x = ±2 where did x = 0 go?
x³ = 4x → x(x − 2)(x + 2) = 0 → x = 0, 2, −2 move everything to one side and factor

Dividing by x silently assumed x ≠ 0 and threw away a genuine zero. In an applied problem there is one more filter that is not algebraic at all: a root meaning a negative length, a negative time, or a fractional object is rejected on meaning. The factoring tells you every zero the polynomial has; only the original problem decides which ones you are allowed to keep.

Graphing from factors, in order. 1. Factor completely — look hard for a difference of squares or a common factor hiding a repeat. 2. Set each factor to zero for the x-intercepts. 3. Read each exponent: odd crosses, even touches; 1 is slanted, 3 or more is flat. 4. Add the multiplicities for the degree, then use the sign of the leading coefficient for the two ends. 5. Plot f(0). 6. Join it up, and check that the number of crossings has the same parity as the degree.

Five ways this goes wrong. • Thinking the exponent moves the zero. (x − 2)³ = 0 still gives x = 2, not 6 or 8. The exponent changes the behaviour there, never the location. • Reading multiplicity from a half-factored form, so a repeated factor like (x² − 4)(x − 2) goes unnoticed. • Calling every horizontal tangent a turning point. Odd multiplicities of 3 or more flatten and then keep going. • Assuming a touch means multiplicity exactly 2. A graph shows parity, not the number. • Forgetting the leading coefficient. Zeros fix the factors but not the vertical scale, and a negative one flips every touch and every crossing direction — use a known point such as f(0) to pin it down.