Complex Zeros of Polynomial Functions Worksheet + Generator
Practice complex zeros, conjugate pairs, multiplicity, and reconstructing real-coefficient polynomials from their zeros. Includes 18 problems, a one-page Conjugate Root Theorem cheat sheet, an interactive polynomial-from-zeros generator, and a complete answer key.
Browse our collection of free polynomial worksheets and answer keys for operations, factoring, division, graphing, and zeros.
Free for classroom, tutoring, homework, AP Precalculus review, and independent practice.
Complex Zeros & Conjugate Root Theorem Cheat Sheet
For a polynomial with real coefficients, non-real zeros come in conjugate pairs. If a + bi is a zero and b ≠ 0, then a - bi must also be a zero, with the same multiplicity.
Download the one-page printable cheat sheet →
Worked example: Missing Polynomial From Its Zeros
Given zeros: 2, -3, 1 + 2i
Because the coefficients are real, add the conjugate 1 - 2i.
(x - 2)(x + 3)(x - (1 + 2i))(x - (1 - 2i))
(x - (1 + 2i))(x - (1 - 2i)) = (x - 1)² + 4 = x² - 2x + 5.
(x - 2)(x + 3)(x² - 2x + 5)
x⁴ - x³ - 3x² + 17x - 30. Degree 4 matches the four zeros counting multiplicity.
Missing Polynomial From Its Zeros Generator
Enter known zeros for a polynomial with real coefficients. The generator adds any missing complex conjugates, preserves multiplicity, builds real factors, and expands the polynomial. It is designed for integer zeros such as 2, -3, 1+2i.
Use commas between zeros. Supported examples: 4, -2, 3+i, 1-2i, 5i, -i. Repeat a zero to show multiplicity. For exact, readable output, each real and imaginary component must be an integer from -10 to 10, and the completed degree is capped at 10.
This generator assumes the polynomial has real coefficients. It constructs a polynomial from the zeros you provide; different leading coefficients produce scalar multiples with the same zeros.
Three Levels of Complex Zeros Practice
Level 1: Conjugates & Degree
Identify the required conjugate zero, match multiplicities, and determine the least possible degree.
Level 2: Build the Polynomial
Start from real and complex zeros, add missing conjugates, combine factors, and write the least-degree monic polynomial.
Level 3: Find All Zeros
Use a known complex zero to obtain its conjugate, divide or factor the remaining polynomial, and finish the complete zero set.
Complex Zeros of Polynomial Functions Worksheet Questions
Complex Zeros Worksheet Answer Key
The key below keeps the central algebra visible on the page. The printable answer-key PDF gives the same results in a grading-friendly format.
| # | Answer | Key factor / reasoning |
|---|---|---|
| 1 | 3 - 4i | Conjugate pair: 3 + 4i and 3 - 4i. |
| 2 | -2 + 5i | Change only the sign of the imaginary part. |
| 3 | -6i | The conjugate of 0 + 6i is 0 - 6i. |
| 4 | 2, -1 + 3i, -1 - 3i; degree 3 | One real zero plus one conjugate pair gives 3 zeros. |
| 5 | -4 (mult. 2), 2 + i, 2 - i; degree 4 | The conjugate has the same multiplicity as its partner; here each complex zero occurs once. |
| 6 | -3, 1 + 2i (mult. 2), 1 - 2i (mult. 2); degree 5 | Conjugate roots have matching multiplicity for real-coefficient polynomials. |
| 7 | x^4 - x^3 - 3x^2 + 17x - 30 | (x - 2)(x + 3)[(x - 1)^2 + 4] = (x - 2)(x + 3)(x^2 - 2x + 5). |
| 8 | x^3 - 7x^2 + 9x + 17 | (x + 1)[(x - 4)^2 + 1] = (x + 1)(x^2 - 8x + 17). |
| 9 | x^3 + 4x^2 + 13x | x[(x + 2)^2 + 9] = x(x^2 + 4x + 13). |
| 10 | x^4 - 4x^3 + 2x^2 - 12x + 45 | (x - 3)^2[(x + 1)^2 + 4] = (x - 3)^2(x^2 + 2x + 5). |
| 11 | x^3 - 2x + 4 | (x + 2)[(x - 1)^2 + 1] = (x + 2)(x^2 - 2x + 2). |
| 12 | x^3 - 5x^2 + 4x - 20 | (x - 5)(x^2 + 4). |
| 13 | 2, 3i, -3i | (x - 2)(x^2 + 9). |
| 14 | -3, 1 + 2i, 1 - 2i | (x + 3)(x^2 - 2x + 5). |
| 15 | -4, 2, i, -i | (x + 4)(x - 2)(x^2 + 1). |
| 16 | 1 (multiplicity 2), -2 + i, -2 - i | (x - 1)^2(x^2 + 4x + 5). |
| 17 | 0, -1, 4, 2 + 3i, 2 - 3i | x(x + 1)(x - 4)(x^2 - 4x + 13). |
| 18 | 2x^5 - 12x^4 + 20x^3 + 40x^2 - 102x + 52 | 2(x + 2)(x - 1)^2[(x - 3)^2 + 4] = 2(x + 2)(x - 1)^2(x^2 - 6x + 13). |
Common Complex Zero & Conjugate Root Mistakes
Forgetting the real-coefficient condition
The conjugate-root requirement applies when the polynomial coefficients are real.
Changing both signs
The conjugate of a + bi is a - bi. The real part stays the same.
Ignoring multiplicity
If a + bi has multiplicity 2, then a - bi also has multiplicity 2 for a real-coefficient polynomial.
Leaving complex coefficients
Multiply conjugate linear factors together first; their product is a quadratic with real coefficients.
Counting only distinct zeros
A degree n polynomial has n complex zeros when multiplicity is counted.
Stopping after the conjugate
When solving a polynomial, use the conjugate pair to factor out a real quadratic and continue with the quotient.
Related Polynomial Worksheets and References
Additional free education references
Created and reviewed by Berke Sahbazoglu
Owner and lead math tutor, Burke Tutoring in Fremont. Berke has 10+ years of tutoring experience and more than 6,000 tutoring hours across K–12 math and science, including extensive Algebra I & II instruction. These problems were adapted from complex-zero practice I use with students and independently checked before publication. This resource was reviewed for conjugate pairing, multiplicity, real-factor construction, polynomial expansion, and final zero sets.
In-Home Tutoring Service Areas
Looking for one-on-one math support beyond this worksheet? Burke Tutoring provides in-home tutoring for students in Fremont, Newark, and Union City.