Polynomial Roots Calculator

This polynomial roots calculator finds the real zeros of any polynomial, from quadratics and cubics to higher degrees. Along with the roots it reports the degree, the leading coefficient, the y-intercept, how many turning points the graph has, and its end behavior (what happens as \( x \to \pm\infty \)). Exact integer and simple fractional roots are shown exactly; others are given to about eight digits.

Use it as a zeros of a polynomial calculator to check factoring, sketch graphs, or find where a function crosses the x-axis. It’s free and needs no sign-up.

How to use the polynomial roots calculator

1. Type the polynomial in x, for example x^3 - 6x^2 + 11x - 6. Use ^ for powers. Implied multiplication such as 2x works.
2. Press Analyze.
3. Read the real roots. If there are fewer real roots than the degree, the rest are repeated or complex, and the note says so.
4. Use the end behavior and turning points to sketch the graph.

Key facts the calculator uses

A polynomial of degree n, \( p(x) = a_nx^n + \dots + a_1x + a_0 \), has:

• At most n real roots. Counting complex roots and multiplicity, it has exactly n (the fundamental theorem of algebra).
• At most \( n - 1 \) turning points, because the derivative has degree \( n - 1 \).
• End behavior set by the leading term \( a_nx^n \):

Degree	\( a_n > 0 \)	\( a_n < 0 \)
Even	up on both ends	down on both ends
Odd	down on the left, up on the right	up on the left, down on the right

Worked example: a cubic with three integer roots

\( p(x) = x^3 - 6x^2 + 11x - 6 \), as loaded above.

Rational root test. Possible rational roots divide the constant term 6: \( \pm1, \pm2, \pm3, \pm6 \). Trying \( x = 1 \): \( 1 - 6 + 11 - 6 = 0 \). So \( (x - 1) \) is a factor.

Divide. \( x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3) \).

Roots. \( x = 1, 2, 3 \). Degree 3 with a positive leading coefficient: falls to \( -\infty \) on the left, rises to \( +\infty \) on the right, with 2 turning points.

Worked example: factor by grouping

\( x^3 - x^2 - 4x + 4 = x^2(x - 1) - 4(x - 1) = (x - 1)(x^2 - 4) = (x - 1)(x - 2)(x + 2) \). The roots are \( -2, 1, 2 \), and the y-intercept is \( (0, 4) \).

Worked example: a quartic with negative leading coefficient

\( p(x) = -2x^4 + 8x^2 = -2x^2(x^2 - 4) \). The roots are \( -2, 0, 2 \), with \( x = 0 \) a double root (the graph touches the axis there instead of crossing). Even degree and \( a_n = -2 < 0 \), so both ends point down.

When roots aren’t nice numbers

\( x^3 - 2x - 5 \) has one real root, \( x \approx 2.0946 \), and two complex roots. No rational root test will find it. The calculator locates it numerically, the same idea as Newton’s method, which you can watch step by step in the Newton’s method calculator.

Tips and common mistakes

• Missing powers. In \( x^3 - 2x - 5 \) the \( x^2 \) coefficient is 0. Keep a 0 placeholder when you do synthetic division.
• Counting double roots once. \( (x - 1)^2 \) contributes two roots at \( x = 1 \), even though the graph touches the axis at only one point.
• Mixing up roots and factors. A root at \( x = -2 \) means the factor is \( (x + 2) \).
• Odd-degree polynomials always have a real root, because the two ends point in opposite directions.

Related guides

• Polynomials: Degree, Roots and End Behavior
• Quadratic Formula
• How to Find Critical Points
• Newton’s Method

Further reading

• Zeroes/Roots of Polynomials (Paul’s Online Math Notes) — multiplicity, the fundamental theorem of algebra and factoring.
• Polynomial (Wikipedia) — definitions, degree, roots and graphs.

FAQ

How do you find the roots of a polynomial?

Try the rational root test, factor out each root you find, and finish the remaining quadratic with the quadratic formula. For roots that aren’t rational, use a numerical method like this calculator does.

How many roots can a polynomial have?

A degree-n polynomial has at most n real roots, and exactly n roots when complex and repeated roots are counted.

What is end behavior?

What \( p(x) \) does as x goes to \( \pm\infty \). It depends only on the degree (even or odd) and the sign of the leading coefficient.

Does it find complex roots?

It reports real roots and tells you how many are repeated or complex. For quadratics, the quadratic formula calculator gives complex roots exactly.