[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-523492-105":59,"doc-detail-523492-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","the-fundamental-theorem-of-algebra-practice-worksheet","The Fundamental Theorem of Algebra - Practice Worksheet","","Kuta Software-Infinite Precalculus worksheet focused on the Fundamental Theorem of Algebra. Students analyze multiple polynomials by stating the number of complex zeros, the possible counts of real and imaginary zeros, the possible numbers of positive and negative real zeros, and the possible rational zeros. The problems then ask for the rational zeros plus an interval containing all real zeros, followed by factoring each polynomial into linear factors and irreducible quadratic factors, and finally finding all zeros.",{"@graph":69,"@context":123},[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/the-fundamental-theorem-of-algebra-practice-worksheet/523492/",{"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/the-fundamental-theorem-of-algebra-practice-worksheet/523492.png","ImageObject",300,407,{"name":92,"@type":93},"nayy☆","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-12","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does the worksheet ask for first for each polynomial?","Question",{"text":113,"@type":114},"It asks for the number of complex zeros and the possible numbers of real and imaginary zeros, plus the possible numbers of positive and negative real zeros and the possible rational zeros.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"After listing rational zeros, what additional step is required?",{"text":118,"@type":114},"Students must give an interval in which all real zeros lie, then factor the polynomial into linear factors and irreducible quadratic factors.",{"name":120,"@type":111,"acceptedAnswer":121},"What does “Find all zeros” problems require?",{"text":122,"@type":114},"They require determining every zero of the given function, including real and complex zeros, based on the polynomial’s structure.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},523492,1791513147,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":52,"preview_page_count":4},962090893153,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Precalculus  \nThe Fundamental Theorem of Algebra  \nName  Date  Period   \nState the number of complex zeros, the possible number of real and imaginary zeros, the possible number of positive and negative zeros, and the possible rational zeros for each function.  \n1) f (x) 􀀠 􀀘x􀀗 􀀐 􀀖􀀙x 􀀕 􀀐 􀀛􀀔 2) f (x) 􀀠 􀀔􀀘 x 􀀘 􀀎 􀀖 x 􀀗 􀀎 􀀔􀀗􀀓x 􀀖 􀀎 􀀕􀀛 x 􀀕 􀀎 􀀗􀀘 x 􀀎 􀀜  \n3) f (x) 􀀠 􀀘x􀀖 􀀐 x 􀀕 􀀐 􀀘x 􀀎 􀀔 4) f (x) 􀀠 􀀖 x 􀀖 􀀎 􀀔􀀔 x 􀀕 􀀎 􀀘x 􀀐 􀀖  \n5) f (x) 􀀠 􀀔􀀓x 􀀘 􀀐 􀀔􀀘 x 􀀗 􀀎 􀀔􀀕x 􀀖 􀀐 􀀔􀀛 x 􀀕 􀀎 􀀕x 􀀐 􀀖 6) f (x) 􀀠 􀀘x􀀘 􀀐 􀀕􀀘 x 􀀗 􀀎 􀀗􀀙x 􀀖 􀀐 􀀕􀀖􀀓x 􀀕 􀀎 􀀜x 􀀐 􀀗􀀘  \nState the possible rational zeros and an interval in which all real zeros lie for each function.  \nThen factor each to linear and irreducible quadratic factors.  \n7) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀔􀀔 x 􀀕 􀀎 􀀜 8) f (x) 􀀠 􀀕􀀚x 􀀖 􀀎 􀀔  \n9) f (x) 􀀠 􀀕x 􀀘 􀀎 􀀔􀀓x 􀀗 􀀐 􀀔􀀖 x 􀀖 􀀐 􀀙􀀘 x 􀀕 􀀐 􀀚x 􀀐 􀀖􀀘 10) f (x) 􀀠 􀀘x􀀘 􀀐 􀀕􀀘 x 􀀗 􀀐 􀀕􀀙x 􀀖 􀀎 􀀔􀀖􀀓x 􀀕 􀀎 􀀘x 􀀐 􀀕􀀘  \n11) f (x) 􀀠 􀀘x􀀗 􀀎 􀀖􀀔 x 􀀕 􀀎 􀀙 12) f (x) 􀀠 􀀕x 􀀗 􀀎 􀀜x 􀀕 􀀎 􀀚  \nFind all zeros.  \n13) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀔􀀜x 􀀕 􀀎 􀀕􀀗 14) f (x) 􀀠 x 􀀖 􀀎 􀀛  \n15) f (x) 􀀠 􀀖 x 􀀘 􀀐 􀀙x 􀀗 􀀎 􀀔􀀗x 􀀖 􀀐 􀀕􀀛 x 􀀕 􀀐 􀀘x 􀀎 􀀔􀀓 16) f (x) 􀀠 􀀘x􀀗 􀀎 􀀙x 􀀕 􀀎 􀀔  \n17) f (x) 􀀠 􀀜x 􀀘 􀀐 􀀔􀀘 x 􀀗 􀀎 􀀘􀀚x􀀖 􀀐 􀀜􀀘 x 􀀕 􀀐 􀀗􀀕x 􀀎 􀀚􀀓 18) f (x) 􀀠 􀀘x􀀗 􀀐 􀀚x􀀕 􀀎 􀀕  \nFactor each to linear factors. One zero has been given.  \n19) f (x) 􀀠 􀀘x􀀘 􀀎 􀀗􀀜x 􀀗 􀀎 􀀔􀀕􀀘 x 􀀖 􀀎 􀀔􀀔􀀖 x 􀀕 􀀎 􀀕􀀕x 􀀐 􀀔􀀓 ; 􀀐􀀗 􀀎 􀀙  \nKuta Software-Infinite Precalculus  \nThe Fundamental Theorem of Algebra  \nName  Date  Period   \nState the number of complex zeros, the possible number of real and imaginary zeros, the possible number of positive and negative zeros, and the possible rational zeros for each function.  \n1) f (x) 􀀠 􀀘x􀀗 􀀐 􀀖􀀙x 􀀕 􀀐 􀀛􀀔  \n\\# of complex zeros: 4  \nPossible \\# of real zeros: 4, 2, or 0  \nPossible \\# of imaginary zeros: 4, 2, or 0 Possible \\# positive real zeros: 1  \nPossible \\# negative real zeros: 1  \nPossible rational zeros:  \n􀁲 􀀔 , 􀁲 􀀖 , 􀁲 􀀜 , 􀁲 􀀕􀀚 , 􀁲 􀀛􀀔 , 􀁲 ~~ ~~􀀔􀀘~~ ~~, 􀁲 ~~ ~~􀀖􀀘~~ ~~, 􀁲 ~~ ~~􀀜􀀘~~ ~~ , 􀁲 􀀕􀀚􀀘~~ ~~, 􀁲 ~~ ~~􀀛􀀔􀀘~~ ~~  \n3) f (x) 􀀠 􀀘x􀀖 􀀐 x 􀀕 􀀐 􀀘x 􀀎 􀀔  \n\\# of complex zeros: 3  \nPossible \\# of real zeros: 3 or 1  \nPossible \\# of imaginary zeros: 2 or 0  \nPossible \\# positive real zeros: 2 or 0 Possible \\# negative real zeros: 1 Possible rational zeros: 􀁲 􀀔 , 􀁲 ~~ ~~􀀔􀀘  \n5) f (x) 􀀠 􀀔􀀓x 􀀘 􀀐 􀀔􀀘 x 􀀗 􀀎 􀀔􀀕x 􀀖 􀀐 􀀔􀀛 x 􀀕 􀀎 􀀕x 􀀐 􀀖  \n\\# of complex zeros: 5  \nPossible \\# of real zeros: 5, 3, or 1  \nPossible \\# of imaginary zeros: 4, 2, or 0 Possible \\# positive real zeros: 5, 3, or 1 Possible \\# negative real zeros: 0  \nPossible rational zeros:  \n􀁲 􀀔 , 􀁲 􀀖 , 􀁲 􀀔􀀕~~ ~~ , 􀁲 􀀖􀀕~~ ~~ , 􀁲 ~~ ~~􀀔􀀘~~ ~~, 􀁲 ~~ ~~􀀖􀀘~~ ~~, 􀁲 ~~ ~~􀀔􀀔􀀓~~ ~~ , 􀁲 ~~ ~~􀀖􀀔􀀓~~ ~~  \n2) f (x) 􀀠 􀀔􀀘 x 􀀘 􀀎 􀀖 x 􀀗 􀀎 􀀔􀀗􀀓x 􀀖 􀀎 􀀕􀀛 x 􀀕 􀀎 􀀗􀀘 x 􀀎 􀀜  \n\\# of complex zeros: 5  \nPossible \\# of real zeros: 5, 3, or 1 Possible \\# of imaginary zeros: 4, 2, or 0 Possible \\# positive real zeros: 0  \nPossible \\# negative real zeros: 5, 3, or 1 Possible rational zeros:  \n􀁲 􀀔 , 􀁲 􀀖 , 􀁲 􀀜 , 􀁲 ~~ ~~􀀔􀀖~~ ~~, 􀁲 ~~ ~~􀀔􀀘~~ ~~, 􀁲 ~~ ~~􀀖􀀘~~ ~~, 􀁲 ~~ ~~􀀜􀀘~~ ~~ , 􀁲 ~~ ~~􀀔􀀔􀀘~~ ~~  \n4) f (x) 􀀠 􀀖 x 􀀖 􀀎 􀀔􀀔 x 􀀕 􀀎 􀀘x 􀀐 􀀖  \n\\# of complex zeros: 3 Possible \\# of real zeros: 3 or 1  \nPossible \\# of imaginary zeros: 2 or 0 Possible \\# positive real zeros: 1 Possible \\# negative real zeros: 2 or 0 Possible rational zeros: 􀁲 􀀔 , 􀁲 􀀖 , 􀁲 ~~ ~~􀀔􀀖~~ ~~  \n6) f (x) 􀀠 􀀘x􀀘 􀀐 􀀕􀀘 x 􀀗 􀀎 􀀗􀀙x 􀀖 􀀐 􀀕􀀖􀀓x 􀀕 􀀎 􀀜x 􀀐 􀀗􀀘  \n\\# of complex zeros: 5  \nPossible \\# of real zeros: 5, 3, or 1 Possible \\# of imaginary zeros: 4, 2, or 0 Possible \\# positive real zeros: 5, 3, or 1 Possible \\# negative real zeros: 0  \nPossible rational zeros:  \n􀁲 􀀔 , 􀁲 􀀖 , 􀁲 􀀘 , 􀁲 􀀜 , 􀁲 􀀔􀀘 , 􀁲 􀀗􀀘 , 􀁲 ~~ ~~􀀔􀀘~~ ~~, 􀁲 ~~ ~~􀀖􀀘~~ ~~, 􀁲 ~~ ~~􀀜􀀘~~ ~~  \nState the possible rational zeros and an interval in which all real zeros lie for each function. Then factor each to linear and irreducible quadratic factors.  \n7) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀔􀀔 x 􀀕 􀀎 􀀜  \nPossible rational zeros:  \n􀁲 􀀔 , 􀁲 􀀖 , 􀁲 􀀜 , 􀁲 􀀔􀀕~~ ~~ , 􀁲 􀀖􀀕~~ ~~ , 􀁲 􀀜􀀕~~  ~~Real zeros lie in: [􀀐􀀖 , 􀀖]  \nFactors to: f (x) 􀀠 (􀀕x 􀀕 􀀐 􀀜)(x 􀀐 􀀔)(x 􀀎 􀀔)  \n8) f (x) 􀀠 􀀕􀀚x 􀀖 􀀎 􀀔  \nPossible rational zeros: 􀁲 􀀔 , 􀁲 ~~ ~~􀀔􀀖~~ ~~, 􀁲 ~~ ~~􀀔􀀜~~ ~~, 􀁲 􀀕􀀔􀀚  \nReal zeros lie in: [􀀐􀀔 , 􀀓]  ","cbCaivqQbfcVeWeD","https://ap.wps.com/l/cbCaivqQbfcVeWeD","pdf",31501,"English","# Problems\n## Complex zero counts and possible real/imaginary zeros\n## Rational zeros, real-zero intervals, and factoring\n## Find all zeros","[{\"question\":\"What does the worksheet ask for first for each polynomial?\",\"answer\":\"It asks for the number of complex zeros and the possible numbers of real and imaginary zeros, plus the possible numbers of positive and negative real zeros and the possible rational zeros.\"},{\"question\":\"After listing rational zeros, what additional step is required?\",\"answer\":\"Students must give an interval in which all real zeros lie, then factor the polynomial into linear factors and irreducible quadratic factors.\"},{\"question\":\"What does “Find all zeros” problems require?\",\"answer\":\"They require determining every zero of the given function, including real and complex zeros, based on the polynomial’s structure.\"}]","The Fundamental Theorem of Algebra - Practice Worksheet | PDF",1791426917]