[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522701-105":59,"doc-detail-522701-en":123},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":116,"head_meta":118,"extra_data":120,"updated_unix":122},105,"en","irrational-and-imaginary-root-theorems-state-the-number-of-complex-zeros-and-possible-realimaginary-zeros","Irrational and Imaginary Root Theorems - State the number of complex zeros and possible real/imaginary zeros","","Worksheet for Infinite Algebra 2 practice focused on irrational and imaginary root theorems for polynomial functions. Students determine the number of complex zeros and the possible counts of real and imaginary zeros for several given polynomials, then identify all additional zeros when some zeros are provided for polynomials with rational coefficients. It also includes tasks to write a least-degree polynomial with integral coefficients that matches specified zeros, plus critical-thinking true/false and construction prompts involving complex roots like 2i and 2.",{"@graph":69,"@context":115},[70,84,106],{"@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/irrational-and-imaginary-root-theorems-state-the-number-of-complex-zeros-and-possible-realimaginary-zeros/522701/",{"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/irrational-and-imaginary-root-theorems-state-the-number-of-complex-zeros-and-possible-realimaginary-zeros/522701.png","ImageObject",300,407,{"name":92,"@type":93},"\tJames","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109],{"name":110,"@type":111,"acceptedAnswer":112},"How can you write a least-degree polynomial with integral coefficients that has given zeros?","Question",{"text":113,"@type":114},"Form the polynomial using the provided real zeros and complex roots (including required conjugates), then expand and simplify to achieve the minimum degree with integral coefficients.","Answer","https://schema.org",{"og:url":83,"og:type":117,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":119,"canonical":83},"index,follow",{"doc_id":121,"site_id":62},522701,1791512073,{"code":4,"msg":5,"data":124},{"doc_id":121,"user_id":125,"nickname":92,"user_avatar":126,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":127,"file_id":128,"file_url":129,"file_type":130,"file_size":131,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":132,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":133,"faqs":134,"seo_title":135,"seo_description":67,"update_tm":136,"read_time":52,"preview_page_count":4},2336474466412,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Kuta Software-Infinite Algebra 2 Name   \nIrrational and Imaginary Root Theorems Date  Period   \nState the number of complex zeros and the possible number of real and imaginary zeros for each function.  \n1) f (x) = x 2 + 6x − 38 2) f (x) = x 4 − 9x2 + 18  \n3) f (x) = 5x5 + 36x3 + 7x 4) f (x) = 3x3 + 2x2 − x  \n5) f (x) = 27x6 + 208x3 − 64 6) f (x) = 27x9 + 8x6 − 27x3 − 8  \nA polynomial function with rational coefficients has the follow zeros. Find all additional zeros.  \n7) −5, i 8) −1 + i, 5  \n9) −3 + 5 , −i 10) 2, −2 + 10  \n11) −1, 5, −2 + 5 12) 2 − 2i, 1 − 2i, 1 + 2i  \n13) 2 − 3i mult. 2 5  \n14) −  2  ,  mult. 2  \nWrite a polynomial function of least degree with integral coefficients that has the given zeros.  \n15) 0, 2, 3 16) −5, 3  \n17) −1, 2i 18) 2i, −2i, 2 + 2i  \n19) −2i, 2 + 2 2 20) 6 , −3 + 5  \nCritical thinking questions:  \n21) Write a polynomial function of fifth degree 22) True or False: A polynomial function of third  \nwith integral coefficients that has 2i as a zero. degree with integral coefficients can have 2  \nand 2i as zeros.  \nKuta Software-Infinite Algebra 2 Name   \nIrrational and Imaginary Root Theorems Date  Period   \nState the number of complex zeros and the possible number of real and imaginary zeros for each function.  \n1) f (x) = x 2 + 6x − 38  \n\\# of complex zeros: 2 Possible \\# of real zeros: 2 or 0  \nPossible \\# of imaginary zeros: 2 or 0  \n3) f (x) = 5x5 + 36x3 + 7x  \n\\# of complex zeros: 5  \nPossible \\# of real zeros: 5, 3, or 1  \nPossible \\# of imaginary zeros: 4, 2, or 0  \n5) f (x) = 27x6 + 208x3 − 64  \n\\# of complex zeros: 6  \nPossible \\# of real zeros: 6, 4, 2, or 0  \nPossible \\# of imaginary zeros: 6, 4, 2, or 0  \n2) f (x) = x 4 − 9x2 + 18  \n\\# of complex zeros: 4  \nPossible \\# of real zeros: 4, 2, or 0  \nPossible \\# of imaginary zeros: 4, 2, or 0  \n4) f (x) = 3x3 + 2x2 − x  \n\\# of complex zeros: 3 Possible \\# of real zeros: 3 or 1  \nPossible \\# of imaginary zeros: 2 or 0  \n6) f (x) = 27x9 + 8x6 − 27x3 − 8  \n\\# of complex zeros: 9  \nPossible \\# of real zeros: 9, 7, 5, 3, or 1  \nPossible \\# of imaginary zeros: 8, 6, 4, 2, or 0  \nA polynomial function with rational coefficients has the follow zeros. Find all additional zeros.  \n7) −5, i  \n−i  \n9) −3 + 5 , −i  \n−3 − 5 , i  \n8) −1 + i, 5  \n−1 − i, − 5  \n10) 2, −2 + 10  \n−2 − 10  \n11) −1, 5, −2 + 5 12) 2 − 2i, 1 − 2i, 1 + 2i  \n−2 − 5 2 + 2i  \n13) 2 − 3i mult. 2 5  \n14) −  2  ,  mult. 2  \n2 + 3i mult. 2  \n− 5 mult. 2  \nWrite a polynomial function of least degree with integral coefficients that has the given zeros.  \n15) 0, 2, 3  \nf (x) = x 4 − 2x3 − 3x2 + 6x  \n17) −1, 2i  \nf (x) = x 3 + x 2 + 4x + 4  \n19) −2i, 2 + 2 2  \nf (x) = x 4 − 4x3 − 16x − 16  \nCritical thinking questions:  \n21) Write a polynomial function of fifth degree with integral coefficients that has 2i as a zero.  \nMany answers. Ex: f (x) = x 5 + 4x3  \n16) −5, 3  \nf (x) = x 3 + 5x2 − 3x − 15  \n18) 2i, −2i, 2 + 2i  \nf (x) = x 4 − 4x3 + 12x2 − 16x + 32  \n20) 6 , −3 + 5  \nf (x) = x 4 + 6x3 − 2x2 − 36x − 24  \n22) True or False: A polynomial function of third degree with integral coefficients can have 2 and 2i as zeros.  \nTrue  \nCreate your own worksheets like this one with Infinite Algebra 2. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaidT57p64NX1K","https://ap.wps.com/l/cbCaidT57p64NX1K","pdf",36314,"English","# State complex zeros and possible real/imaginary zeros\n## Given polynomial functions\n# Find additional zeros for rational-coefficient polynomials\n## Provided zeros (including conjugates)\n# Write least-degree polynomials with integral coefficients\n## Match given zeros\n# Critical thinking questions","[{\"question\":\"How can you write a least-degree polynomial with integral coefficients that has given zeros?\",\"answer\":\"Form the polynomial using the provided real zeros and complex roots (including required conjugates), then expand and simplify to achieve the minimum degree with integral coefficients.\"}]","Irrational and Imaginary Root Theorems - State the number of complex zeros and possible real/imaginary zeros | PDF",1791426074]