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Students evaluate f(x) at given k values, determine the remainder when f(x) is divided by x−k, and analyze upper and lower bound test results to decide whether k is an upper bound, lower bound, or neither for real zeros. Additional problems ask for an interval containing all real zeros.",{"@graph":69,"@context":127},[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-remainder-theorem-and-bounds-kuta-software-infinite-precalculus/523493/",{"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-remainder-theorem-and-bounds-kuta-software-infinite-precalculus/523493.png","ImageObject",300,407,{"name":92,"@type":93},"nayy☆","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",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119,123],{"name":110,"@type":111,"acceptedAnswer":112},"What does the worksheet ask students to do with f(x) and k?","Question",{"text":113,"@type":114},"It asks students to evaluate f(x) at specified k values and use those results in later remainder and bound questions.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the remainder found in the worksheet problems?",{"text":118,"@type":114},"The worksheet directs students to find the remainder when f(x) is divided by x - k.",{"name":120,"@type":111,"acceptedAnswer":121},"How do the upper and lower bound tests determine k?",{"text":122,"@type":114},"Students use the upper and lower bound test outcomes to classify k as an upper bound, a lower bound, or neither bound for the real zeros of f(x).",{"name":124,"@type":111,"acceptedAnswer":125},"What is the final type of problem on the worksheet?",{"text":126,"@type":114},"Students determine an interval in which all real zeros of f(x) lie.","https://schema.org",{"og:url":83,"og:type":129,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":131,"canonical":83},"index,follow",{"doc_id":133,"site_id":62},523493,1791557047,{"code":4,"msg":5,"data":136},{"doc_id":133,"user_id":137,"nickname":92,"user_avatar":138,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":139,"file_id":140,"file_url":141,"file_type":142,"file_size":143,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":144,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":145,"faqs":146,"seo_title":147,"seo_description":67,"update_tm":148,"read_time":52,"preview_page_count":4},962090893153,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Precalculus Name   \nThe Remainder Theorem and Bounds Date  Period   \nEvaluate f (x) at k.  \n1) f (x) 􀀠 x 􀀖 􀀐 􀀗x 􀀕 􀀎 􀀚x 2) f (x) 􀀠 x 􀀖 􀀐 􀀔􀀙x k 􀀠 􀀐􀀔 k 􀀠 􀀔  \n3) f (x) 􀀠 x 􀀗 􀀎 􀀗x 􀀖 􀀎 􀀘x􀀕 􀀐 􀀗x 􀀐 􀀗 4) f (x) 􀀠 􀀘x􀀘 􀀐 􀀗x 􀀗 􀀐 􀀘x􀀕 􀀐 􀀘 k 􀀠 􀀔 k 􀀠 􀀔  \nFind the remainder when f (x) is divided by x 􀀐 k.  \n5) f (x) 􀀠 􀀘x􀀙 􀀎 􀀘x􀀘 􀀐 􀀚x􀀗 􀀎 􀀚x􀀖 􀀎 􀀘x􀀕 􀀐 􀀗x 􀀐 􀀔 6) f (x) 􀀠 􀀘x􀀗 􀀐 􀀙x 􀀖 􀀐 x 􀀕 􀀎 􀀚x 􀀐 􀀕 k 􀀠 􀀐􀀕 k 􀀠 􀀕  \n7) f (x) 􀀠 x 􀀗 􀀎 􀀔􀀕x 􀀖 􀀎 􀀖􀀚x 􀀕 􀀎 􀀗􀀕x 􀀎 􀀔􀀙 8) f (x) 􀀠 􀀗x 􀀙 􀀐 􀀘x􀀘 􀀐 􀀜x 􀀗 􀀎 􀀕x 􀀕 􀀎 􀀖 x 􀀐 􀀜 k 􀀠 􀀐􀀖 k 􀀠 􀀐􀀔  \nDetermine whether the upper and lower bound tests indicate k is an upper bound, a lower bound, or neither bound on the real zeros of f (x).  \n9) f (x) 􀀠 x 􀀘 􀀎 􀀘x􀀗 􀀎 􀀘x􀀖 􀀎 x 􀀕 􀀐 􀀖 x 􀀐 􀀗 10) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀕x 􀀖 􀀐 􀀙x 􀀕 􀀎 􀀙x k 􀀠 􀀗 k 􀀠 􀀐􀀔  \n11) f (x) 􀀠 􀀘x􀀘 􀀐 􀀖 x 􀀗 􀀎 􀀘x􀀖 􀀎 􀀗x 􀀕 􀀎 􀀘x 12) f (x) 􀀠 x 􀀘 􀀐 􀀘x􀀗 􀀎 􀀔􀀕x 􀀖 􀀎 􀀘x􀀕 􀀐 􀀔􀀖 x k 􀀠 􀀗 k 􀀠 􀀐􀀘  \n13) f (x) 􀀠 x 􀀗 􀀎 􀀗x 􀀖 􀀐 􀀘x􀀕 􀀐 􀀘x 14) f (x) 􀀠 􀀕x 􀀙 􀀎 􀀘x􀀘 􀀎 􀀗x 􀀗 􀀎 􀀖 x 􀀖 􀀎 􀀗x 􀀕 􀀎 􀀖 x 􀀐 􀀘 k 􀀠 􀀖 k 􀀠 􀀐􀀖  \nFind an interval in which all real zeros of f (x) lie.  \n15) f (x) 􀀠 􀀔􀀕x 􀀗 􀀐 􀀕􀀘 x 􀀖 􀀐 􀀛 x 􀀕 􀀎 􀀕􀀖 x 􀀐 􀀙 16) f (x) 􀀠 x 􀀘 􀀐 􀀖 x 􀀗 􀀐 x 􀀖 􀀎 􀀖 x 􀀕 􀀎 􀀖 x 􀀎 􀀔  \n17) f (x) 􀀠 􀀖 x 􀀗 􀀐 􀀛 x 􀀖 􀀐 􀀔􀀘 x 􀀕 􀀐 􀀗x 18) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀜x 􀀖 􀀐 􀀚x􀀕 􀀎 􀀜x 􀀎 􀀘  \nKuta Software-Infinite Precalculus The Remainder Theorem and Bounds Evaluate f (x) at k.  \n1) f (x) 􀀠 x 􀀖 􀀐 􀀗x 􀀕 􀀎 􀀚x k 􀀠 􀀐􀀔  \n􀀐􀀔􀀕  \n3) f (x) 􀀠 x 􀀗 􀀎 􀀗x 􀀖 􀀎 􀀘x􀀕 􀀐 􀀗x 􀀐 􀀗 k 􀀠 􀀔  \n􀀕  \nFind the remainder when f (x) is divided by x 􀀐 k.  \n5) f (x) 􀀠 􀀘x􀀙 􀀎 􀀘x􀀘 􀀐 􀀚x􀀗 􀀎 􀀚x􀀖 􀀎 􀀘x􀀕 􀀐 􀀗x 􀀐 􀀔 k 􀀠 􀀐􀀕  \n􀀔􀀜  \n7) f (x) 􀀠 x 􀀗 􀀎 􀀔􀀕x 􀀖 􀀎 􀀖􀀚x 􀀕 􀀎 􀀗􀀕x 􀀎 􀀔􀀙 k 􀀠 􀀐􀀖  \n􀀐􀀕􀀓  \nName  Date  Period   \n2) f (x) 􀀠 x 􀀖 􀀐 􀀔􀀙x k 􀀠 􀀔  \n􀀐􀀔􀀘  \n4) f (x) 􀀠 􀀘x􀀘 􀀐 􀀗x 􀀗 􀀐 􀀘x􀀕 􀀐 􀀘 k 􀀠 􀀔  \n􀀐􀀜  \n6) f (x) 􀀠 􀀘x􀀗 􀀐 􀀙x 􀀖 􀀐 x 􀀕 􀀎 􀀚x 􀀐 􀀕 k 􀀠 􀀕  \n􀀗􀀓  \n8) f (x) 􀀠 􀀗x 􀀙 􀀐 􀀘x􀀘 􀀐 􀀜x 􀀗 􀀎 􀀕x 􀀕 􀀎 􀀖 x 􀀐 􀀜 k 􀀠 􀀐􀀔  \n􀀐􀀔􀀓  \nDetermine whether the upper and lower bound tests indicate k is an upper bound, a lower bound, or neither bound on the real zeros of f (x).  \n9) f (x) 􀀠 x 􀀘 􀀎 􀀘x􀀗 􀀎 􀀘x􀀖 􀀎 x 􀀕 􀀐 􀀖 x 􀀐 􀀗 10) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀕x 􀀖 􀀐 􀀙x 􀀕 􀀎 􀀙x  \nk 􀀠 􀀗 k 􀀠 􀀐􀀔  \nUpper bound Neither  \n11) f (x) 􀀠 􀀘x􀀘 􀀐 􀀖 x 􀀗 􀀎 􀀘x􀀖 􀀎 􀀗x 􀀕 􀀎 􀀘x 12) f (x) 􀀠 x 􀀘 􀀐 􀀘x􀀗 􀀎 􀀔􀀕x 􀀖 􀀎 􀀘x􀀕 􀀐 􀀔􀀖 x k 􀀠 􀀗 k 􀀠 􀀐􀀘  \nUpper bound Lower bound  \n13) f (x) 􀀠 x 􀀗 􀀎 􀀗x 􀀖 􀀐 􀀘x􀀕 􀀐 􀀘x 14) f (x) 􀀠 􀀕x 􀀙 􀀎 􀀘x􀀘 􀀎 􀀗x 􀀗 􀀎 􀀖 x 􀀖 􀀎 􀀗x 􀀕 􀀎 􀀖 x 􀀐 􀀘 k 􀀠 􀀖 k 􀀠 􀀐􀀖  \nUpper bound Lower bound  \nFind an interval in which all real zeros of f (x) lie.  \n15) f (x) 􀀠 􀀔􀀕x 􀀗 􀀐 􀀕􀀘 x 􀀖 􀀐 􀀛 x 􀀕 􀀎 􀀕􀀖 x 􀀐 􀀙 16) f (x) 􀀠 x 􀀘 􀀐 􀀖 x 􀀗 􀀐 x 􀀖 􀀎 􀀖 x 􀀕 􀀎 􀀖 x 􀀎 􀀔  \n[􀀐􀀔 , 􀀖] [􀀐􀀔 , 􀀗]  \n17) f (x) 􀀠 􀀖 x 􀀗 􀀐 􀀛 x 􀀖 􀀐 􀀔􀀘 x 􀀕 􀀐 􀀗x 18) f (x) 􀀠 􀀕x 􀀗 􀀐 􀀜x 􀀖 􀀐 􀀚x􀀕 􀀎 􀀜x 􀀎 􀀘  \n[􀀐􀀕 , 􀀗] [􀀐􀀕 , 􀀙]  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaidM3vl7aVs1G","https://ap.wps.com/l/cbCaidM3vl7aVs1G","pdf",27752,"English","# Evaluate f(x) at k\n# Find the remainder when f(x) is divided by x - k\n# Upper and lower bound tests for real zeros\n# Interval containing all real zeros","[{\"question\":\"What does the worksheet ask students to do with f(x) and k?\",\"answer\":\"It asks students to evaluate f(x) at specified k values and use those results in later remainder and bound questions.\"},{\"question\":\"How is the remainder found in the worksheet problems?\",\"answer\":\"The worksheet directs students to find the remainder when f(x) is divided by x - k.\"},{\"question\":\"How do the upper and lower bound tests determine k?\",\"answer\":\"Students use the upper and lower bound test outcomes to classify k as an upper bound, a lower bound, or neither bound for the real zeros of f(x).\"},{\"question\":\"What is the final type of problem on the worksheet?\",\"answer\":\"Students determine an interval in which all real zeros of f(x) lie.\"}]","The Remainder Theorem and Bounds - Kuta Software - Infinite Precalculus | PDF",1791426920]