torised"}]},"order":3,"title":"Cube one below","description":"Now expand the binomial for an integer one less than a multiple of 3. The bracket is an integer, so this is one less than a multiple of 9. [A1]"},{"type":"hyper_latex","block":{"nodes":[{"latex":"(3n+1)^3=\\htmlData{anchor=expand_plus}{\\color{@green}{27n^3+27n^2+9n+1}}"},{"latex":"27n^3+27n^2+9n+1=\\htmlData{anchor=factor_plus}{\\color{@green}{9(3n^3+3n^2+n)+1}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"expand_plus"},{"color":"green","style":"text-color","anchor":"factor_plus"}]},"order":5,"title":"Cube one above","description":"Finally, expand the binomial for an integer one more than a multiple of 3. The bracket is an integer, so this is one more than a multiple of 9. [A1] Since the three forms 3n, 3n-1 and 3n+1 cover every integer, every cube is a multiple of 9, one less than a multiple of 9, or one more than a multiple of 9."}]}],"generatedAt":"2026-07-21T04:10:06.094Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"c9224f41-b6a2-4b50-bebe-3e3ac01d4329","revisionId":"d40fe4a6-5fe9-4df7-bcad-c62bc823fc1c","questionNumber":"29","questionType":"LA","specification":"$$g(x) = 2x^3 + x^2 - 38x + 35$","difficulty":1,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"parts":[{"id":"8086eed7-11ae-4cba-9af4-80813057d5e6","label":"a","content":"Use the factor theorem to show that $(x + 5)$ is a factor of $g(x)$.","markscheme":"- Evaluate $g(-5) = 2(-5)^3 + (-5)^2 - 38(-5) + 35 = -250 + 25 + 190 + 35 = 0$. M1\n- Conclude $(x+5)$ is a factor. A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempts to evaluate $g(-5)$.","code":"M"},{"id":"p2","marks":1,"description":"Shows $g(-5) = 0$ and concludes $(x+5)$ is a factor.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.612Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"factor_theorem","label":"Factor theorem","steps":[{"type":"text","order":0,"title":"Choose the test value","content":"For a possible factor $(x+5)$, use the factor theorem with $x=-5$. We now test whether $g(-5)=0$.","highlights":[{"text":"Use the factor theorem to show that $(x + 5)$ is a factor of $g(x)$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"g(-5)=\\htmlData{anchor=substitute}{\\color{@sky}{2(-5)^3+(-5)^2-38(-5)+35}}"},{"latex":"g(-5)=\\htmlData{anchor=values}{\\color{@sky}{-250+25+190+35}}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{g(-5)=0}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"values","from":"substitute","color":"sky","label":"evaluate powers and products","style":"solid"}]},"order":1,"title":"Evaluate the function","description":"Substitute -5 carefully, keeping brackets around the negative number. This complete correct evaluation earns the M1."},{"type":"text","order":2,"title":"State the conclusion","content":"Since $g(-5)=0$, the factor theorem tells us that $(x-(-5))=(x+5)$ is a factor of $g(x)$. Stating this conclusion earns **A1**."}]}],"generatedAt":"2026-07-21T04:01:46.980Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"1f042d36-2268-4aa9-8d11-788d7f99a04f","label":"b","content":"Hence, show that $g(x)$ can be written as the product of three linear factors.","markscheme":"- Divide $g(x)$ by $(x+5)$ to get $2x^2 - 9x + 7$. M1 A1\n- Factorise quadratic: $(2x-7)(x-1)$. M1\n- State final form: $(x+5)(2x-7)(x-1)$. A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":2,"description":"Finds the quadratic factor $2x^2 - 9x + 7$.","code":"M"},{"id":"p2","marks":1,"description":"Factorises the quadratic into $(2x-7)(x-1)$.","code":"M"},{"id":"p3","marks":1,"description":"States the product of thr

Algebraic Methods

Question 25

Algebraic Methods Questions

Algebraic Methods

Question 25

Algebraic Methods Questions