A zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.
In Spanish
cero de una función
Also called
zero, root of a function, zeros of a function, solution of f(x) = 0
Each factor gives one zero. Where f(x) = 0, the graph is on the x-axis.
Formal definition and notation
A zero of a functionf is an element a of its domain such that f(a) = 0. For a polynomial, a is a zero exactly when (x − a) is a factor (the factor theorem). Real zeros correspond to the x-intercepts(a, 0) of the graph; a polynomial may also have complex zeros, which do not appear on the graph.
Symbol
Read it as
Meaning
f(a) = 0
f of a equals zero
a is a zero of f
(x − a)
x minus a
the factor that goes with the zero a
x = ± 3
x equals plus or minus 3
a short way to write two zeros, 3 and −3
Key facts
Zero, root, solution and x-intercept are linked: the zero 3 of f is a root of f(x) = 0, a solution of that equation, and gives the x-intercept (3, 0).
A polynomial of degreen has at most n real zeros, so a quadratic has at most 2.
A repeated factor such as (x − 1)2 gives a zero where the graph touches the axis and turns back instead of crossing.
A function can have no real zeros: f(x) = x2 + 1 is always at least 1.
The zero of a linear functionf(x) = mx + b with m ≠ 0 is −bm.
Zeros from factored form
Zeros from factored form
If f(x) = a(x − r1)(x − r2)⋯, the zeros are r1, r2, …
a
a nonzero constant; it changes the shape but not the zeros
r1, r2
the zeros: set each factor equal to 0 (zero product property)
Watch the signs: the factor (x + 2) gives the zero −2.
The number 0 is the value f reaches. A zero of a function is the input where that happens, and it is usually not 0 itself.
Quick check: 3 questions
Question 1 of 3
What are the zeros of h(x) = (x − 7)(x + 4)?
Show answer and explanation
Answer: B.x − 7 = 0 gives 7, and x + 4 = 0 gives −4. The first choice copies the signs from the factors, −28 is h(0), and 7 and 4 drops a sign.
Question 2 of 3
Which function has no real zeros?
Show answer and explanation
Answer: D.x2 + 4 ≥ 4 for every real x, so it never equals 0. x2 − 4 has zeros ± 2, x2 has the zero 0, and 4 − x has the zero 4.
Question 3 of 3
A function’s graph crosses the x-axis at (−1, 0) and (6, 0). Which could be the function?
Show answer and explanation
Answer: A. Zeros −1 and 6 come from the factors (x + 1) and (x − 6). The second choice has the zeros 1 and −6, the third has ± √6, and the last has −1 and −6.
Zero of a Function: frequently asked questions
What is the difference between a zero and a root?
They name the same number from two points of view. A zero belongs to a function: f(a) = 0. A root belongs to an equation: a solves f(x) = 0. Teachers and tests use the words almost interchangeably, so “find the zeros of f” and “find the roots of f(x) = 0” ask for the same values.
How do you find the zeros of a function?
Set the function equal to 0 and solve. For a linear function, isolate x. For a quadratic, factor and use the zero product property, or use the quadratic formula when it will not factor. For a graph, read the x-values where the curve meets the x-axis. Then check each zero by substituting it back in.
Can a zero of a function be 0?
Yes. If f(0) = 0, then 0 is a zero and the graph passes through the origin. For example, f(x) = x2 − 5x = x(x − 5) has zeros 0 and 5. The word zero describes the output, which is always 0; the input that produces it can be any number, including 0.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-APR.B.3, CCSS HSF-IF.C.8.a
Cite this entry
Effortless Math Team. “Zero of a Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/zero-of-a-function/
Flashcard
TermZero of a Functionf(a) = 0
DefinitionAn input that makes the output 0; where the graph meets the x-axis. f(x) = (x − 2)(x + 5) has zeros 2 and −5.
FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2X-InterceptAn x-intercept is a point where a graph crosses or touches the x-axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1, Algebra 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Difference of SquaresA difference of squares is an expression of the form a2 − b2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x2 − 49 = (x + 7)(x − 7). A sum of squares, a2 + b2, does not factor over the real numbers.Algebra · Algebra 1, Algebra 2HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Function NotationFunction notation is the way of writing a function as f(x), read “f of x”: f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1, Algebra 2Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusSolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8