Polynomials and Conjugate Roots - Exercises - Find Additional Zeros

Oct 8, 2026

Worksheets for Infinite Precalculus on polynomial functions with rational or integral coefficients, focusing on zeros and conjugate root relationships. Problems ask students to identify all additional zeros implied by given rational roots and complex conjugate pairs, including cases with multiplicity. Additional tasks require constructing the lowest-degree polynomial with integer coefficients that has specified zeros. A critical-thinking section explains why certain polynomial degrees guarantee extra real roots and prompts writing a degree-ten polynomial with two imaginary roots.

LanguageEnglish

CategoryExam

FormatPDF

Pages2

Reading Time5 min

AccessFree to download

Polynomials and Conjugate Roots

Find all additional zeros

Write least-degree polynomial with integer coefficients

Critical thinking questions

Kuta Software-Infinite Precalculus Name Polynomials and Conjugate Roots Date Period A polynomial function with rational coefficients has the follow zeros. Find all additional zeros. 1) 􀀐􀀔 , 􀀔 􀀎 􀀖 i 2) 􀀐 ~~ ~~􀀔􀀗~~ ~~ , 􀀔 􀀎 􀀙 3) 􀀐􀀖 mult. 2, 􀀕 􀀕 4) 􀀔 􀀎 􀀖 , 􀀐􀀖 􀀎 􀀘 5) 􀀔 􀀐 i, 6) 􀀐􀀖 􀀎 􀀕 i, 􀀐􀀕 􀀐 􀀕 i, 􀀐􀀕 􀀎 􀀕 i Write a polynomial function of least degree with integral coefficients that has the given zeros. 7) 􀀐 􀀔 􀀔 􀀖 8) 􀀐􀀔 , 􀀐i 􀀕 , , 􀀗 9) 􀀕 mult. 3 10) 􀀐􀀖 , 􀀕 􀀕 11) 􀀐􀀖 , 􀀖 12) 􀀔 􀀎 􀀔􀀓 mult. 2, 􀀔 􀀐 􀀔􀀓 13) 􀀐i mult. 2 14) 􀀗􀀘~~ ~~ , 􀀕 i Critical thinking questions: 15) Explain why it makes sense that a 16) Write a polynomial function of degree ten third-degree polynomial must have at least that has two imaginary roots. one rational zero. Kuta Software-Infinite Precalculus Name Polynomials and Conjugate Roots Date Period A polynomial function with rational coefficients has the follow zeros. Find all additional zeros. 1) 􀀐􀀔 , 􀀔 􀀎 􀀖 i 􀀔 2) 􀀐 ~~ ~~ , 􀀔 􀀎 􀀙 􀀔 􀀐 􀀖 i 􀀗 􀀔 􀀐 􀀙 3) 􀀐􀀖 mult. 2, 􀀕 􀀕 4) 􀀔 􀀎 􀀖 , 􀀐􀀖 􀀎 􀀘 􀀐􀀕 􀀕 􀀔 􀀐 􀀖 , 􀀐􀀖 􀀐 􀀘 5) 􀀔 􀀐 i, 6) 􀀐􀀖 􀀎 􀀕 i, 􀀐􀀕 􀀐 􀀕 i, 􀀐􀀕 􀀎 􀀕 i 􀀔 􀀎 i, 􀀐 􀀚 􀀐􀀖 􀀐 􀀕 i Write a polynomial function of least degree with integral coefficients that has the given zeros. 7) 􀀐 ~~ ~~􀀔􀀕~~ ~~, 􀀔 , 􀀖􀀗~~ ~~f (x) 􀀠 􀀛 x 􀀖 􀀐 􀀔􀀓x 􀀕 􀀐 x 􀀎 􀀖 9) 􀀕 mult. 3 f (x) 􀀠 x 􀀖 􀀐 􀀙x 􀀕 􀀎 􀀔􀀕x 􀀐 􀀛 8) 􀀐􀀔 , 􀀐i f (x) 􀀠 x 􀀖 􀀎 x 􀀕 􀀎 x 􀀎 􀀔 10) 􀀐􀀖 , 􀀕 􀀕 f (x) 􀀠 x 􀀖 􀀎 􀀖 x 􀀕 􀀐 􀀛 x 􀀐 􀀕􀀗 11) 􀀐􀀖 , 􀀖 12) 􀀔 􀀎 􀀔􀀓 mult. 2, 􀀔 􀀐 􀀔􀀓 f (x) 􀀠 x 􀀖 􀀎 􀀖 x 􀀕 􀀐 􀀖 x 􀀐 􀀜 f (x) 􀀠 x 􀀗 􀀐 􀀗x 􀀖 􀀐 􀀔􀀗x 􀀕 􀀎 􀀖􀀙x 􀀎 􀀛􀀔 13) 􀀐i mult. 2 14) 􀀗 􀀕 if (x) 􀀠 x 􀀗 􀀎 􀀕x 􀀕 􀀎 􀀔 􀀘 , f (x) 􀀠 􀀘x􀀖 􀀐 􀀗x 􀀕 􀀎 􀀕􀀓x 􀀐 􀀔􀀙 Critical thinking questions: 15) Explain why it makes sense that a 16) Write a polynomial function of degree ten third-degree polynomial must have at least that has two imaginary roots. one ratIt musonal zerogo from .􀁦 to 􀀐􀁦 so it must f (x) 􀀠 (x 􀀕 􀀎 􀀔)􀀘 cross the x-axis. Create your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)

FAQs

Why do complex zeros imply additional zeros for polynomials with rational coefficients?

Because non-real zeros of such polynomials occur in conjugate pairs, the given complex zeros force additional conjugate zeros.

How do you write a polynomial of least degree with integer coefficients from given zeros?

Use the zeros to build factors, using the conjugate pairing when complex roots appear, then combine into the smallest-degree polynomial consistent with all listed zeros and their multiplicities.

What does the critical thinking question about a third-degree polynomial’s rational zero mean?

It focuses on the idea that a cubic with a rational zero must produce a real root scenario that makes the graph cross the x-axis, connecting rational roots to real-intercept behavior.

Oct 8, 2026

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