Zeros of a Function - Definition, Methods, Examples

TL;DR

A zero of a function f is any input value x for which f(x)=0. On a graph, the zeros are the x-coordinates where the curve crosses or touches the x-axis. This article covers the formal definition, the four standard methods for finding zeros, multiplicity of zeros, the three common errors, and the historical story that connects zeros to the rise of polynomial equations.

The Problem That Forced Mathematicians To Ask "for which x does this vanish?"

In 1545, Italian mathematician Gerolamo Cardano published Ars Magna — a book that solved cubic equations for the first time in print. The cubic he opened with was x³ + 6x = 20, and the question wasn't what shape does it have? but for which value of x does the left side equal the right? In modern terms: find the zeros of f(x)=x³ + 6x−20. The hunt for zeros is the oldest question in algebra.

A zero of a function f is a value x=c such that f(c)=0. The same value is also called a root of the equation f(x)=0 and an x-intercept of the graph y=f(x). Three names, one object — pick whichever vocabulary the question uses.

Why "zero," "root," and "x-intercept" All Mean The Same Thing

• Zero of a function. Used when the focus is on the function f.
• Root of an equation. Used when the focus is on the equation f(x)=0.
• x-intercept of a graph. Used when the focus is on the curve y=f(x).

A polynomial of degree n has at most n real zeros — the Fundamental Theorem of Algebra says it has exactly n complex zeros (counted with multiplicity). For Grade 9–12 work, "zero" almost always means real zero unless complex roots are explicitly asked for.

Four Methods To Find Zeros

Method 1 — Factoring

Set f(x)=0. Factor the left side. Use the zero-product property: if a product of factors equals zero, at least one factor must be zero. Set each factor to zero and solve.

Method 2 — The Quadratic Formula

When factoring isn't clean — coefficients ugly or roots irrational — use:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression b²−4ac is the discriminant. If positive, two real zeros. If zero, one real zero (a double root). If negative, no real zeros (two complex zeros).

Method 3 — Graphical Inspection

Graph y=f(x) on Desmos, GeoGebra, or by hand. The x-coordinates of the points where the curve crosses or touches the x-axis are the zeros. Works for every function — gives approximate values, not exact ones.

Method 4 — Rational Root Theorem (for polynomials of degree ≥ 3)

For a polynomial with integer coefficients aₙxⁿ + ... + a₀, any rational zero p/q has p dividing a₀ and q dividing aₙ. Test each candidate with synthetic division. Once one zero is found, divide out the corresponding factor and reduce to a lower-degree polynomial.

Quick — Standard — Stretch: three worked examples

Quick — find the zeros of f(x)=x²−9

Set x²−9=0, so x²=9, giving x=±3.

Final answer: Zeros are x=3 and x=−3.

Standard (Wrong-Path-First) — find the zeros of f(x)=x²+4x+13

Wrong path. Try factoring first. Look for two numbers that multiply to 13 and add to 4. There are no integer factors. The wrong path stops here and declares "no zeros."

Correct method. Apply the quadratic formula. a=1, b=4, c=13.

x = \frac{-4 \pm \sqrt{16 - 52}}{2} = \frac{-4 \pm \sqrt{-36}}{2} = -2 \pm 3i

Final answer: Zeros are x=−2+3i and x=−2−3i — two complex conjugate zeros, no real zeros.

Stretch — find the zeros of f(x)=x³−4x²+x+6

Try the rational root theorem. a₀=6, aₙ=1. Candidates: ±1,±2,±3,±6.

Test x=−1: (−1)³−4(−1)²+(−1)+6=0. ✓ So x=−1 is a zero.

Final answer: Zeros are x=−1, x=2, and x=3 — three real zeros, matching the degree of the cubic.

Multiplicity — When A Zero Shows Up More Than Once

A zero can repeat. The factor (x−2)² in a polynomial gives x=2 as a zero with multiplicity 2. On a graph, the curve touches the x-axis at that zero rather than crossing it.

Why Zeros Matter — From Antenna Design To Drug Dosing

Zeros tell you the input values where a function vanishes — which translates directly into the moments and conditions that matter in applied work.

Mistakes Students Make When Finding Zeros

Mistake 1: Declaring "no solution" when the discriminant is negative

Correct way: Compute the discriminant b²−4ac first. If negative, the function has two complex zeros.

Mistake 2: Forgetting that a zero can have multiplicity > 1

Correct way: Count with multiplicity; the cubic above has three zeros counted with multiplicity.

Mistake 3: Forgetting to check the candidate roots from the rational root theorem

Correct way: Test each candidate by substitution or synthetic division.

Routing Table for Methods to Find Zeros

Key Takeaways

• A zero of a function is an input x=c for which f(c)=0.
• The four standard methods are factoring, the quadratic formula, graphical inspection, and the rational root theorem.
• The discriminant tells you the kind of zeros a quadratic has before you compute them.
• Every domain that asks "when does this quantity hit zero?" is asking for the zeros of a function.

Try these — three problems

1. Find the zeros of f(x)=x²−7x+12.
2. Find the zeros of f(x)=x²+6x+13. State whether they are real or complex.
3. Find all zeros of f(x)=x³−2x²−5x+6 using the rational root theorem.