Polynomial Calculator
Enter a polynomial's coefficients and a value for x below to instantly evaluate it.
Evaluating a Polynomial
Substitute the given value in for x and simplify each term, then add everything together.
A polynomial's degree — its highest exponent — determines its end behavior. 2x³ − 3x² + 4x − 5 is degree 3 (a cubic), so its two ends point in opposite directions: it falls toward negative infinity on the left and rises toward positive infinity on the right.
Worked example
Evaluate 2x³ − 3x² + 4x − 5 at x = 2.
2(8) − 3(4) + 4(2) − 5 = 16 − 12 + 8 − 5
Result: 7
A second example, with a negative x: evaluate the same polynomial at x = −1.
2(−1) − 3(1) + 4(−1) − 5 = −2 − 3 − 4 − 5
Result: −14
Common mistake
When plugging in a negative x, it's easy to forget that the sign of each term depends on whether its exponent is even or odd. (−1)² = 1, but (−1)³ = −1 — losing track of that flips the sign of an entire term and throws off the final result.
Keep going
- Evaluating a polynomial at a value and dividing it by (x − c) are closely related operations; the synthetic division calculator shows the full division, not just the evaluated result. Synthetic Division Calculator
Frequently Asked Questions
What is a polynomial?
A polynomial is an expression built from variables raised to whole-number powers, multiplied by coefficients, and added together — like 2x³ − 3x² + 4x − 5.
How do you evaluate a polynomial at a value?
Substitute the value in for x everywhere it appears and simplify. This calculator uses Horner's method, which evaluates a polynomial with fewer multiplications than plugging in term by term.
What order should the coefficients be entered in?
Highest degree first. For 2x³ − 3x² + 4x − 5, enter 2, −3, 4, −5.
What is the maximum number of real roots a polynomial can have?
A polynomial of degree n can have at most n real roots — values of x where the polynomial equals 0. A cubic like 2x³ − 3x² + 4x − 5 can have up to 3 real roots, though it may have fewer if some roots are complex.
What determines a polynomial's end behavior?
Its degree and leading coefficient. An odd-degree polynomial with a positive leading coefficient falls on the left and rises on the right; an even-degree polynomial with a positive leading coefficient rises on both ends.
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