The formula this page uses
Any rational zero p/q of aₙxⁿ + … + a₀ has p dividing a₀ and q dividing aₙ
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Zeros Of A Polynomial Calculator
Solve zeros of a polynomial problems with clear steps, notation, and a final check.
x²
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Algebra
Zeros Of A Polynomial Calculator explained
The short version
- A zero is an x value that makes the polynomial equal 0. On a graph it is where the curve meets the x-axis.
- The rational root theorem builds a short list of candidates: factors of the constant over factors of the leading coefficient.
- A degree-n polynomial has exactly n zeros once complex numbers are allowed, but only the real ones show up on the graph.
What each part means
SymbolWhat it means
a₀ — Constant termIts whole-number factors supply every possible numerator p.
aₙ — Leading coefficientIts factors supply every possible denominator q.
p/q — Candidate zeroTest each one. A remainder of 0 means it is a genuine zero.
multiplicity — How many times a zero repeatsA zero of multiplicity 2 touches the axis and bounces instead of crossing.
Show your work: a full example
- The polynomialf(x) = 2x³ − 3x² − 8x − 3
- Factors of the constant −3 give p±1, ±3
- Factors of the leading 2 give q±1, ±2
- Build the candidate list p/q±1, ±3, ±1/2, ±3/2
- Test x = −12(−1)³ − 3(−1)² − 8(−1) − 3 = −2 − 3 + 8 − 3 = 0, so x = −1 is a zero
- Divide out (x + 1) by synthetic divisioncoefficients 2, −3, −8, −3 reduce to 2, −5, −3 with remainder 0, so the quotient is 2x² − 5x − 3
- Factor the quotient2x² − 5x − 3 = (2x + 1)(x − 3), giving the remaining zeros x = −1/2 and x = 3
A second, different case
- A different case: complex zerosg(x) = x³ − x² + 4x − 4
- Group the terms in pairs(x³ − x²) + (4x − 4)
- Factor each pairx²(x − 1) + 4(x − 1)
- Pull out the shared bracket(x − 1)(x² + 4)
- First zerox − 1 = 0, so x = 1
- Second factorx² + 4 = 0, so x² = −4 and x = 2i or x = −2i
- Read the graphthree zeros in total but only one is real, so the curve crosses the x-axis exactly once
Copy-ready example
f(x) = 2x³ − 3x² − 8x − 3
The polynomial
Questions about the Zeros Of A Polynomial Calculator
What is the difference between a zero, a root and an x-intercept?
A zero and a root are the same thing said about a function and about an equation. An x-intercept is the graph's version, and it only exists for real zeros, which is why 2i is a root of x³ − x² + 4x − 4 but never shows up on the picture.
Why does the rational root theorem never find √2?
Because it only searches fractions, and √2 cannot be written as one whole number over another. For x² − 2 the candidate list is ±1, ±2, all of which fail, which is itself proof that the zeros are irrational.
What does a repeated zero look like on a graph?
An even multiplicity makes the curve touch the axis and turn back, like y = (x − 3)² at x = 3. An odd multiplicity above 1 makes it flatten out as it passes through, like y = (x − 3)³.
How many zeros does a degree-n polynomial have?
Exactly n, counting complex zeros and counting repeats. That is the fundamental theorem of algebra. The degree-3 examples above each have three, even though one shows three crossings and the other shows one.