[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-208561-en":3,"doc-seo-208561-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},208561,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",4,"Exam","Worksheet 10 - Factor Theorem - AQA Level 2 Further Mathematics","Worksheet 10 on the Factor Theorem for AQA Level 2 Certificate Further Mathematics (8365). The questions guide students through polynomial factorisation, identifying zeros where graphs cross the x-axis, and using the factor theorem to find values of parameters and linear factors. It includes problems finding linear factors from given polynomials, determining constants in cubic expressions, and solving cubic and quintic equations using established factor results and full factorisation.","|  | AQA Qualifications |  |\n| --- | --- | --- |\n|  |  |  |\n| AQA Level 2 Certificate\u003Cbr>FURTHER MATHEMATICS\u003Cbr>Level 2 (8365) |  |  |\n| Worksheet 10\u003Cbr>Factor Theorem |  |  |\n\nOur specification is published on our website ([www.aqa.org.uk](www.aqa.org.uk) ) . We will let centres know in writing about any changes to the specification. We will also publish changes on our website. The definitive version of our specification will always be the one on our website, this may differ from printed versions.  \nYou can download this resource from our All About Maths website ([http://allaboutmaths.aqa.org.uk/](http://allaboutmaths.aqa.org.uk/)) .  \nCopyright © 2018 AQA and its licensors. All rights reserved.  \nAQA retains the copyright on all its publications, including the specifications. However, registered centres for AQA are permitted to copy material from this specification booklet for their own internal use.  \nAQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723) . Our registered address is AQA, Devas Street, Manchester M15 6EX.  \n10 Factor Theorem  \nQuestion 1  \n(a) Show that x(x + 4)(x 􀀐 9) = x 3 􀀐 5x2 􀀐 36x (1 mark)  \n(b) Write down the x values of the three points where the graph of y = x 3 􀀐 5x2 􀀐 36x  \ncrosses the x-axis. (2 marks)  \nQuestion 2  \nf(x) = x 3 + 2x2 􀀐 5x 􀀐 6  \n(a) Work out f(1) and f(􀀐1) (2 marks)  \n(b) Work out f(2) and f(􀀐2) (2 marks)  \n(c) Work out f(3) and f(􀀐3) (2 marks)  \n(d) Write down the three linear factors of f(x) . (1 mark)  \nQuestion 3  \n(a) Show that (x + 5) is a factor of x 3 + 7x2 + 2x 􀀐 40 (2 marks)  \n(b) Work out the other two linear factors of x 3 + 7x2 + 2x 􀀐 40 (3 marks)  \n(c) Hence, solve x 3 + 7x2 + 2x 􀀐 40 = 0 (1 mark)  \nQuestion 4  \nA sketch of y = x 3 + 5x2 + 9x + k where k is an integer, is shown.  \n􀀐2 x  \nWork out the value of k. (3 marks)  \nQuestion 5  \n(a) (x + 3) is a factor of f(x) = x 3 + x 2 + ax 􀀐 72 where a is an integer.  \nWork out the value of a. (3 marks)  \n(b) Work out the other linear factors of f(x) . (3 marks)  \nQuestion 6  \n(x − 3) and (x + 4) are factors of f(x) = x 3 + ax 2 + bx + 24 where a and b are integers.  \n(a) Work out the third linear factor of f(x) . (2 marks)  \n(b) Work out the values of a and b. (4 marks)  \nQuestion 7  \n(a) (x − 5) is a factor of f(x) = x3 + kx2 + 9x − 20 where k is an integer.  \nWork out the value of k. (3 marks)  \n(b) Express f(x) as a product of (x − 5) and a quadratic factor. (2 marks)  \n(c) Show that (x − 5) is the only linear factor of f(x) . (2 marks)  \nLEVEL 2 CERTIFICATE FURTHER MATHEMATICS  \nQuestion 8  \nSolve x 3 􀀐 6x2 􀀐 25x − 18 = 0 (5 marks)  \nQuestion 9  \nf(x) = x 5 􀀐 2x4 􀀐 81x + 162 = 0  \n(a) Use the factor theorem to show that f(x) has a factor of (x – 2)  \n(1 mark)  \n(b) Hence work out the integer solutions of f(x) = 0  \n(4 marks)  \nQuestion 10  \n(a) Use the factor theorem to show that (3x + 2) is a factor of 3x3 + 2x2 – 3x – 2  \n(2 marks)  \n(b) Factorise fully 3x3 + 2x2 – 3x – 2  \n(2 marks)","cbCaijp5cqLVzLEQ","https://ap.wps.com/l/cbCaijp5cqLVzLEQ","pdf",841989,1,5,"English","en",105,"# Worksheet 10: Factor Theorem\n## Question 1\n## Question 2\n## Question 3\n## Question 4\n## Question 5\n## Question 6\n## Question 7\n## Question 8\n## Question 9\n## Question 10","[{\"question\":\"What topics does Worksheet 10: Factor Theorem cover?\",\"answer\":\"It covers using the factor theorem to find factors and linear factors of polynomials, parameter determination in cubic expressions, and solving polynomial equations using factorisation results.\"},{\"question\":\"How do questions use the factor theorem in this worksheet?\",\"answer\":\"Several questions ask students to show that a given expression is a factor (e.g., x+5, x−5, 3x+2) and then use that factor to deduce solutions or complete factorisation.\"},{\"question\":\"What types of answers are expected for the different questions?\",\"answer\":\"Students typically compute function values at specific x-values, write down x-intercepts (zeros), determine constants such as a, b, and k, and provide full factorisations and equation solutions.\"}]","Worksheet 10 - Factor Theorem - AQA Level 2 Further Mathematics | PDF",1788604561,13,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"worksheet-10-factor-theorem-aqa-level-2-further-mathematics","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/worksheet-10-factor-theorem-aqa-level-2-further-mathematics/208561/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What topics does Worksheet 10: Factor Theorem cover?","Question",{"text":74,"@type":75},"It covers using the factor theorem to find factors and linear factors of polynomials, parameter determination in cubic expressions, and solving polynomial equations using factorisation results.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do questions use the factor theorem in this worksheet?",{"text":79,"@type":75},"Several questions ask students to show that a given expression is a factor (e.g., x+5, x−5, 3x+2) and then use that factor to deduce solutions or complete factorisation.",{"name":81,"@type":72,"acceptedAnswer":82},"What types of answers are expected for the different questions?",{"text":83,"@type":75},"Students typically compute function values at specific x-values, write down x-intercepts (zeros), determine constants such as a, b, and k, and provide full factorisations and equation solutions.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,103,107,112,117,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":101,"slug":102},70,"exam",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},"Comic",60,"comic",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},6,"Technology",50,"technology",{"id":113,"doc_module":4,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},8,"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":21,"slug":137},19,"General","general"]