Analyze polynomial factors from one entered complex zero. Reveal conjugates, quotient pieces, and remaining roots. Practice smarter factoring with graphs, exports, and worked output.
Enter coefficients in descending powers. Use commas or spaces between values.
| Polynomial coefficients | Entered zero | Conjugate-pair factor | Final factorization |
|---|---|---|---|
| 1, -3, 1, 7, -30 | 1 + 2i | x^2 - 2x + 5 | (x^2 - 2x + 5)(x - 3)(x + 2) |
| 1, 0, 5, 0, 4 | 0 + 1i | x^2 + 1 | (x^2 + 1)(x^2 + 4) |
| 2, -8, 30, -16, 52 | 2 + 3i | x^2 - 4x + 13 | 2(x^2 - 4x + 13)(x^2 + 2) |
For real-coefficient polynomials, nonreal zeros arrive in conjugate pairs. If a + bi is a zero, then a - bi is also a zero.
The paired zeros create this real quadratic factor:
(x - (a + bi))(x - (a - bi)) = x^2 - 2ax + (a^2 + b^2)
After building that quadratic, divide the original polynomial by it. The quotient can then be factored again using real roots or additional complex-conjugate pairs.
Polynomials with real coefficients always pair nonreal roots as conjugates. If one zero is a + bi, the matching zero is a - bi. Their pair produces a real quadratic factor.
Enter coefficients from the highest degree down to the constant. For x^4 - 3x^3 + x^2 + 7x - 30, enter 1,-3,1,7,-30.
Yes. Decimal coefficients and decimal zero parts are accepted. The calculator validates the zero numerically, so small rounding differences are usually handled well.
The residual measures how closely your entered zero fits the polynomial. Smaller values are better. A large residual suggests a typing error or a zero that does not belong to that polynomial.
The conjugate pair multiplies into a quadratic with real coefficients. That real factor is easier to divide out, leaving a smaller polynomial to factor afterward.
Yes. After removing the conjugate-pair factor, the tool solves the quotient numerically and reports the remaining roots and the reconstructed factorization.
Yes. It is useful for verifying a known complex zero, checking the conjugate theorem, and confirming the final factored form before you submit your work.
The Plotly graph maps computed zeros on the complex plane. Real roots sit on the horizontal axis, while nonreal roots appear above and below it.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.