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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.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/polynomials-and-conjugate-roots-exercises-find-additional-zeros/523489/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/polynomials-and-conjugate-roots-exercises-find-additional-zeros/523489.png","ImageObject",300,407,{"name":92,"@type":93},"PakDamar76","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why do complex zeros imply additional zeros for polynomials with rational coefficients?","Question",{"text":112,"@type":113},"Because non-real zeros of such polynomials occur in conjugate pairs, the given complex zeros force additional conjugate zeros.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you write a polynomial of least degree with integer coefficients from given zeros?",{"text":117,"@type":113},"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.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the critical thinking question about a third-degree polynomial’s rational zero mean?",{"text":121,"@type":113},"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.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},523489,1791661258,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":24,"preview_page_count":4},962090893057,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Kuta Software-Infinite Precalculus Name   \nPolynomials and Conjugate Roots Date  Period   \nA polynomial function with rational coefficients has the follow zeros. Find all additional zeros.  \n1) 􀀐􀀔 , 􀀔 􀀎 􀀖 i 2) 􀀐 ~~ ~~􀀔􀀗~~ ~~ , 􀀔 􀀎 􀀙  \n3) 􀀐􀀖 mult. 2, 􀀕 􀀕 4) 􀀔 􀀎 􀀖 , 􀀐􀀖 􀀎 􀀘  \n5) 􀀔 􀀐 i,  6) 􀀐􀀖 􀀎 􀀕 i, 􀀐􀀕 􀀐 􀀕 i, 􀀐􀀕 􀀎 􀀕 i  \nWrite a polynomial function of least degree with integral coefficients that has the given zeros.  \n7) 􀀐  􀀔  􀀔  􀀖  8) 􀀐􀀔 , 􀀐i  \n􀀕 , , 􀀗  \n9) 􀀕 mult. 3 10) 􀀐􀀖 , 􀀕 􀀕  \n11) 􀀐􀀖 , 􀀖 12) 􀀔 􀀎 􀀔􀀓 mult. 2, 􀀔 􀀐 􀀔􀀓  \n13) 􀀐i mult. 2 14) 􀀗􀀘~~ ~~ , 􀀕 i  \nCritical thinking questions:  \n15) Explain why it makes sense that a 16) Write a polynomial function of degree ten  \nthird-degree polynomial must have at least that has two imaginary roots.  \none rational zero.  \nKuta Software-Infinite Precalculus Name   \nPolynomials and Conjugate Roots Date  Period   \nA polynomial function with rational coefficients has the follow zeros. Find all additional zeros.  \n1) 􀀐􀀔 , 􀀔 􀀎 􀀖 i 􀀔  \n2) 􀀐 ~~ ~~ , 􀀔 􀀎 􀀙  \n􀀔 􀀐 􀀖 i 􀀗  \n􀀔 􀀐 􀀙   \n3) 􀀐􀀖 mult. 2, 􀀕 􀀕 4) 􀀔 􀀎 􀀖 , 􀀐􀀖 􀀎 􀀘  \n􀀐􀀕 􀀕 􀀔 􀀐 􀀖 , 􀀐􀀖 􀀐 􀀘  \n5) 􀀔 􀀐 i,  6) 􀀐􀀖 􀀎 􀀕 i, 􀀐􀀕 􀀐 􀀕 i, 􀀐􀀕 􀀎 􀀕 i  \n􀀔 􀀎 i, 􀀐 􀀚 􀀐􀀖 􀀐 􀀕 i  \nWrite a polynomial function of least degree with integral coefficients that has the given zeros.  \n7) 􀀐 ~~ ~~􀀔􀀕~~ ~~, 􀀔 , 􀀖􀀗~~  ~~f (x) 􀀠 􀀛 x 􀀖 􀀐 􀀔􀀓x 􀀕 􀀐 x 􀀎 􀀖  \n9) 􀀕 mult. 3  \nf (x) 􀀠 x 􀀖 􀀐 􀀙x 􀀕 􀀎 􀀔􀀕x 􀀐 􀀛  \n8) 􀀐􀀔 , 􀀐i  \nf (x) 􀀠 x 􀀖 􀀎 x 􀀕 􀀎 x 􀀎 􀀔  \n10) 􀀐􀀖 , 􀀕 􀀕  \nf (x) 􀀠 x 􀀖 􀀎 􀀖 x 􀀕 􀀐 􀀛 x 􀀐 􀀕􀀗  \n11) 􀀐􀀖 , 􀀖 12) 􀀔 􀀎 􀀔􀀓 mult. 2, 􀀔 􀀐 􀀔􀀓  \nf (x) 􀀠 x 􀀖 􀀎 􀀖 x 􀀕 􀀐 􀀖 x 􀀐 􀀜 f (x) 􀀠 x 􀀗 􀀐 􀀗x 􀀖 􀀐 􀀔􀀗x 􀀕 􀀎 􀀖􀀙x 􀀎 􀀛􀀔  \n13) 􀀐i mult. 2 14) 􀀗  􀀕 if (x) 􀀠 x 􀀗 􀀎 􀀕x 􀀕 􀀎 􀀔 􀀘 ,  \nf (x) 􀀠 􀀘x􀀖 􀀐 􀀗x 􀀕 􀀎 􀀕􀀓x 􀀐 􀀔􀀙  \nCritical thinking questions:  \n15) Explain why it makes sense that a 16) Write a polynomial function of degree ten  \nthird-degree polynomial must have at least that has two imaginary roots.  \none ratIt musonal zerogo from .􀁦 to 􀀐􀁦 so it must f (x) 􀀠 (x 􀀕 􀀎 􀀔)􀀘  \ncross the x-axis.  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaijz0Wkh4WPJy","https://ap.wps.com/l/cbCaijz0Wkh4WPJy","pdf",23954,"English","# Polynomials and Conjugate Roots\n## Find all additional zeros\n## Write least-degree polynomial with integer coefficients\n## Critical thinking questions","[{\"question\":\"Why do complex zeros imply additional zeros for polynomials with rational coefficients?\",\"answer\":\"Because non-real zeros of such polynomials occur in conjugate pairs, the given complex zeros force additional conjugate zeros.\"},{\"question\":\"How do you write a polynomial of least degree with integer coefficients from given zeros?\",\"answer\":\"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.\"},{\"question\":\"What does the critical thinking question about a third-degree polynomial’s rational zero mean?\",\"answer\":\"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.\"}]","Polynomials and Conjugate Roots - Exercises - Find Additional Zeros | PDF",1791426912]