[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522750-105":59,"doc-detail-522750-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","the-rational-root-theorem-worksheet-2","The Rational Root Theorem - Worksheet 2","","Algebra 2 worksheet focused on the Rational Root Theorem. Students identify all possible rational zeros for multiple polynomials by listing candidate factors based on the constant term and leading coefficient, then determine the actual rational zeros for each function. The set includes both tasks that only require possible zeros and tasks that require complete solution sets. A critical-thinking prompt explains whether previously tested candidates must be retested after finding a new zero.",{"@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/the-rational-root-theorem-worksheet-2/522750/",{"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-rational-root-theorem-worksheet-2/522750.png","ImageObject",300,407,{"name":92,"@type":93},"Kyle","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},"How do you determine the possible rational zeros for a polynomial using the Rational Root Theorem?","Question",{"text":112,"@type":113},"List candidate fractions formed from factors of the constant term over factors of the leading coefficient, including both positive and negative values. Then compare candidates with the polynomial to see which actually work.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the difference between finding possible rational zeros and finding all rational zeros?",{"text":117,"@type":113},"Possible rational zeros are candidate answers generated by the theorem. All rational zeros are the subset of those candidates that actually satisfy the equation f(x)=0, including multiplicities.",{"name":119,"@type":110,"acceptedAnswer":120},"After finding that a new value is a rational zero, do you need to retest the earlier candidates from the same polynomial? Why or why not?",{"text":121,"@type":113},"No. Once you establish that the earlier candidates are not zeros, the polynomial cannot suddenly have them as zeros. Factoring reasoning shows that if the original polynomial does not include an earlier factor, its factors cannot introduce that missing factor.","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},522750,1791586194,{"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":19,"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":52,"preview_page_count":4},3985741905716,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Kuta Software-Infinite Algebra 2 Name___________________________________  \nThe Rational Root Theorem Date________________ Period____  \nState the possible rational zeros for each function.  \n1) f (x) = 3x2 + 2x − 1 2) f (x) = x 6 − 64  \n3) f (x) = x2 + 8x + 10 4) f (x) = 5x3 − 2x2 + 20x − 8  \n5) f (x) = 4x5 − 2x4 + 30x3 − 15x2 + 50x − 25 6) f (x) = 5x4 + 32x2 − 21  \n7) f (x) = x 3 − 27 8) f (x) = 2x4 − 9x2 + 7  \nState the possible rational zeros for each function. Then find all rational zeros.  \n9) f (x) = x 3 + x2 − 5x + 3 10) f (x) = x 3 − 13x2 + 23x − 11  \n11) f (x) = x 3 + 4x2 + 5x + 2 12) f (x) = 5x3 + 29x2 + 19x − 5  \n13) f (x) = 4x3 − 9x2 + 6x − 1 14) f (x) = 3x3 + 11x2 + 5x − 3  \n15) f (x) = 5x4 − 46x3 + 84x2 − 50x + 7 16) f (x) = 3x4 − 10x3 − 24x2 − 6x + 5  \n17) f (x) = 3x3 + 9x2 + 4x + 12 18) f (x) = 2x3 + 9x2 + 19x + 15  \nCritical thinking question:  \n19) In the process of solving 2x3 + 7x2 + 9x + 10 = 0 you test 1, 2, 5, and 10 as possible zeros and determine  \n5  \nthat none of them are actual zeros. You then discover that − is a zero. You calculate the depressed  \n2  \npolynomial to be 2x3 + 2x + 4. Do you need to test 1, 2, 5, and 10 again? Why or why not?  \nKuta Software-Infinite Algebra 2  \nThe Rational Root Theorem  \nState the possible rational zeros for each function.  \n1) f (x) = 3x2 + 2x − 1  \n± 1, ± ~~ ~~13  \nName___________________________________  \nDate________________ Period____  \n2) f (x) = x 6 − 64  \n± 1, ± 2, ± 4, ± 8, ± 16, ± 32, ± 64  \n3) f (x) = x2 + 8x + 10 4) f (x) = 5x3 − 2x2 + 20x − 8  \n± 1, ± 2, ± 5, ± 10 1 2 4 8  \n± 1, ± 2, ± 4, ± 8, ±  5  , ±  5  , ±  5  , ±  5  \n5) f (x) = 4x5 − 2x4 + 30x3 − 15x2 + 50x − 25 6) f (x) = 5x4 + 32x2 − 21  \n± 1, ± 5, ± 25, ± ~~ ~~12~~ ~~ , ± ~~ ~~52~~ ~~ , ± ~~ ~~252~~ ~~ , ± ~~ ~~14~~ ~~ , ± ~~ ~~54~~ ~~ , ± ~~ ~~254~~ ~~ ± 1, ± 3, ± 7, ± 21, ± ~~ ~~15~~ ~~ , ± ~~ ~~35~~ ~~ , ± ~~ ~~75~~ ~~ , ± ~~ ~~215  \n7) f (x) = x 3 − 27 8) f (x) = 2x4 − 9x2 + 7  \n± 1, ± 3, ± 9, ± 27 1 7  \n± 1, ± 7, ±  2  , ±  2  \nState the possible rational zeros for each function. Then find all rational zeros.  \n9) f (x) = x 3 + x2 − 5x + 3  \nPossible rational zeros: ± 1 , ± 3  \nRational zeros: {−3, 1 mult. 2}  \n10) f (x) = x 3 − 13x2 + 23x − 11  \nPossible rational zeros: ± 1 , ± 11  \nRational zeros: {1 mult. 2, 11}  \n11) f (x) = x 3 + 4x2 + 5x + 2  \nPossible rational zeros: ± 1 , ± 2  \nRational zeros: {−1 mult. 2, −2}  \n13) f (x) = 4x3 − 9x2 + 6x − 1  \nPossible rational zeros: ± 1 , ± ~~ ~~12~~ ~~ , ± ~~ ~~14  \nRational zeros: { 1 mult. 2, 14~~ ~~ }  \n15) f (x) = 5x4 − 46x3 + 84x2 − 50x + 7  \nPossible rational zeros: ± 1 , ± 7, ± ~~ ~~15~~ ~~ , ± ~~ ~~75  \nRational zeros: {~~ ~~15~~ ~~ , 7, 1 mult. 2}  \n17) f (x) = 3x3 + 9x2 + 4x + 12  \nPossible rational zeros:  \n± 1, ± 2, ± 3, ± 4, ± 6, ± 12, ± ~~ ~~13~~ ~~ , ± ~~ ~~23~~ ~~ , ± ~~ ~~43 Rational zeros: {−3}  \n12) f (x) = 5x3 + 29x2 + 19x − 5  \nPossible rational zeros: ± 1 , ± 5, ± ~~ ~~15  \nRational zeros: {~~ ~~15~~ ~~ , −5, −1 }  \n14) f (x) = 3x3 + 11x2 + 5x − 3  \nPossible rational zeros: ± 1 , ± 3, ± ~~ ~~13  \nRational zeros: {~~ ~~13~~ ~~ , −3, −1 }  \n16) f (x) = 3x4 − 10x3 − 24x2 − 6x + 5  \nPossible rational zeros: ± 1 , ± 5, ± ~~ ~~13~~ ~~ , ± ~~ ~~53  \nRational zeros: {~~ ~~13~~ ~~ , 5, −1 mult. 2}  \n18) f (x) = 2x3 + 9x2 + 19x + 15  \nPossible rational zeros:  \n± 1, ± 3, ± 5, ± 15, ± ~~ ~~12~~ ~~ , ± ~~ ~~32~~ ~~ , ± ~~ ~~52~~ ~~ , ± ~~ ~~152  \nRational zeros: {− ~~ ~~32~~ ~~ }  \nCritical thinking question:  \n19) In the process of solving 2x3 + 7x2 + 9x + 10 = 0 you test 1, 2, 5, and 10 as possible zeros and determine  \n5  \nthat none of them are actual zeros. You then discover that − is a zero. You calculate the depressed  \n2  \npolynomial to be 2x3 + 2x + 4. Do you need to test 1, 2, 5, and 10 again? Why or why not?  \nNo. That would be like factoring 740 and discovering 3 isn't a factor but then checking if  \nanything 740 breaks down into has a factor of 3. If the original problem doesn't have a  \nfactor of 3 then nothing it factor","cbCaieXQkWGNwCQz","https://ap.wps.com/l/cbCaieXQkWGNwCQz","pdf",37789,"English","# State the possible rational zeros\n## List candidate rational zeros for each polynomial\n# Find all rational zeros\n## Determine actual rational zeros for selected functions\n# Critical thinking question\n## Retesting candidate zeros after discovering a new zero","[{\"question\":\"How do you determine the possible rational zeros for a polynomial using the Rational Root Theorem?\",\"answer\":\"List candidate fractions formed from factors of the constant term over factors of the leading coefficient, including both positive and negative values. Then compare candidates with the polynomial to see which actually work.\"},{\"question\":\"What is the difference between finding possible rational zeros and finding all rational zeros?\",\"answer\":\"Possible rational zeros are candidate answers generated by the theorem. All rational zeros are the subset of those candidates that actually satisfy the equation f(x)=0, including multiplicities.\"},{\"question\":\"After finding that a new value is a rational zero, do you need to retest the earlier candidates from the same polynomial? Why or why not?\",\"answer\":\"No. Once you establish that the earlier candidates are not zeros, the polynomial cannot suddenly have them as zeros. Factoring reasoning shows that if the original polynomial does not include an earlier factor, its factors cannot introduce that missing factor.\"}]","The Rational Root Theorem - Worksheet 2 | PDF",1791426124]