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bracket to $9x^2 + 12x + 4$.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the quadratic by $(2x - 1)$.","code":"M"},{"id":"p3","marks":1,"description":"Fully simplifies to reach the target expression $18x^3 + 15x^2 - 4x - 4$.","code":"A"}],"finalAnswers":["18x^3 + 15x^2 - 4x - 4"],"generatedAt":"2026-05-07T02:01:01.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"361fff12-8aa1-4e7c-8bb6-1f071bed0bcc","revisionId":"c603963f-e6a5-47b4-8a53-0522477cb323","questionNumber":"13","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Expanding Triple Brackets","calculatorUse":null,"isCalculation":true,"parts":[{"id":"9f5937aa-76c1-411d-970c-4887f1db76c9","label":"0","content":"Show that $(3x - 1)(4x + 3)(x - 9) = 12x^3 - 103x^2 - 48x + 27$ for all values of $x$.","markscheme":"- Expands two brackets correctly, e.g., $(4x^2 - 33x - 27)$ M1\n- Multiplies by $(3x - 1)$ M1\n- Completes expansion to show the required expression $12x^3 - 103x^2 - 48x + 27$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly expands any two brackets to a quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the quadratic by the remaining linear bracket.","code":"M"},{"id":"p3","marks":1,"description":"Fully simplifies to reach the target expression $12x^3 - 103x^2 - 48x + 27$.","code":"A"}],"finalAnswers":["12x^3 - 103x^2 - 48x + 27"],"generatedAt":"2026-05-07T02:01:01.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"successive_expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Choose two brackets","content":"Start with the two brackets that make a quadratic: $$(4x+3)(x-9).$$\n\nExpand by multiplying **each** term in the first bracket by each term in the second. This is the first method mark: a correct expansion of these two brackets earns **M1**.","highlights":[{"text":"$$(3x - 1)(4x + 3)(x - 9)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(4x+3)(x-9)=4x^2-36x+3x-27"},{"latex":"(4x+3)(x-9)=4x^2+\\htmlData{anchor=combine_x}{(-36x+3x)}-27"},{"latex":"(4x+3)(x-9)=4x^2+\\htmlData{anchor=quadratic_x}{(-33x)}-27"}],"connections":[{"to":"quadratic_x","from":"combine_x","color":"sky","label":"combine like terms"}]},"order":1,"title":"Expand the first pair","description":"Keep signs attached: the two products with negative nine are both negative."},{"type":"text","order":2,"title":"Use the remaining bracket","content":"Substitute the quadratic back into the original product, leaving the remaining factor outside:\n\n$$(3x-1)(4x+3)(x-9)=(3x-1)(4x^2-33x-27).$$\n\nWriting this multiplication earns the second **M1**. Now multiply the quadratic by both $3x$ and $-1$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(3x-1)(4x^2-33x-27)"},{"latex":"=\\htmlData{anchor=three_x}{3x}(4x^2-33x-27)-\\htmlData{anchor=one}{1}(4x^2-33x-27)"},{"latex":"=12x^3-99x^2-81x-4x^2+33x+27"},{"latex":"=12x^3-103x^2-48x+27"}],"highlights":[{"color":"green","style":"text-color","anchor":"three_x"},{"color":"orange","style":"text-color","anchor":"one"}]},"order":3,"title":"Multiply every term","description":"Distribute both terms in the first bracket across all three terms of the quadratic."},{"type":"text","order":4,"title":"Compare the result","content":"The final line is exactly the expression required:\n\n$$\\boxed{(3x-1)(4x+3)(x-9)=12x^3-103x^2-48x+27}.$$\n\nTherefore the identity is true **for all values of $x$**. Matching and stating the required cubic completes the **A1**."}]}],"generatedAt":"2026-09-01T09:10:34.247Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9f5937aa-76c1-411d-970c-4887f1db76c9:successive_expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Choose two brackets","content":"Start with the two brackets that make a quadratic: $$(4x+3)(x-9).$$\n\nExpand by multiplying **each** term in the first bracket by each term in the second. This is the first method mark: a correct expansion of these two brackets earns **M1**.","highlights":[{"text":"$$(3x - 1)(4x + 3)(x - 9)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(4x+3)(x-9)=4x^2-36x+3x-27"},{"latex":"(4x+3)(x-9)=4x^2+\\htmlData{anchor=combine_x}{(-36x+3x)}-27"},{"latex":"(4x+3)(x-9)=4x^2+\\htmlData{anchor=quadratic_x}{(-33x)}-27"}],"connections":[{"to":"quadratic_x","from":"combine_x","color":"sky","label":"combine like terms"}]},"order":1,"title":"Expand the first pair","description":"Keep signs attached: the two products with negative nine are both negative."},{"type":"text","order":2,"title":"Use the remaining bracket","content":"Substitute the quadratic back into the original product, leaving the remaining factor outside:\n\n$$(3x-1)(4x+3)(x-9)=(3x-1)(4x^2-33x-27).$$\n\nWriting this multiplication earns the second **M1**. Now multiply the quadratic by both $3x$ and $-1$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(3x-1)(4x^2-33x-27)"},{"latex":"=\\htmlData{anchor=three_x}{3x}(4x^2-33x-27)-\\htmlData{anchor=one}{1}(4x^2-33x-27)"},{"latex":"=12x^3-99x^2-81x-4x^2+33x+27"},{"latex":"=12x^3-103x^2-48x+27"}],"highlights":[{"color":"green","style":"text-color","anchor":"three_x"},{"color":"orange","style":"text-color","anchor":"one"}]},"order":3,"title":"Multiply every term","description":"Distribute both terms in the first bracket across all three terms of the quadratic."},{"type":"text","order":4,"title":"Compare the result","content":"The final line is exactly the expression required:\n\n$$\\boxed{(3x-1)(4x+3)(x-9)=12x^3-103x^2-48x+27}.$$\n\nTherefore the identity is true **for all values of $x$**. Matching and stating the required cubic completes the **A1**."}]}],"generatedAt":"2026-09-01T09:10:34.576Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"4d6510f2-d4bb-4bce-bfd8-290ae3662f18","revisionId":"28299233-183c-4580-b787-94cf3aee97bf","questionNumber":"14","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Expanding Triple Brackets","calculatorUse":null,"isCalculation":true,"parts":[{"id":"22ba0d8f-4fd9-462c-903a-4acf2c01a8b2","label":"0","content":"Show that $(5x - 4)(3x + 1)(2x - 7) = 30x^3 - 119x^2 + 41x + 28$ for all values of $x$.","markscheme":"- Expands two brackets correctly, e.g., $(6x^2 - 19x - 7)$ M1\n- Multiplies by $(5x - 4)$ M1\n- Completes expansion to show the required expression $30x^3 - 119x^2 + 41x + 28$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly expands any two brackets to a quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the quadratic by the remaining linear bracket.","code":"M"},{"id":"p3","marks":1,"description":"Fully simplifies to reach the target expression $30x^3 - 119x^2 + 41x + 28$.","code":"A"}],"finalAnswers":["30x^3 - 119x^2 + 41x + 28"],"generatedAt":"2026-05-07T02:01:01.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths","summaries":[{"id":"64df4672-97ef-49d3-b8f4-76fec52a2de0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4bc64585-6eb9-4a8f-b764-ab56a3d6130e","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"78f75685-fcb2-4bff-a809-54d392f9efea","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"354889bd-222b-4947-b54b-528370558cd9","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0fd7b290-efd6-4943-96c7-0856f8590076","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8fe72371-59f3-44b6-9690-13451b80b444","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bb22e8bb-f94a-4614-a71f-aeabd147d1d5","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"43fccfe1-09e4-4c0c-91d8-d4a6d3232863","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2fdef4b0-1217-4b8d-aba4-596aa5cc5d97","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88d03f80-e249-4b37-a6f7-b0e553166a40","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ca392bc2-3d0b-4ba7-b0d5-81d9b5790693","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9f293ed4-1f33-4831-83e2-f7ac1340982c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"361fff12-8aa1-4e7c-8bb6-1f071bed0bcc","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4d6510f2-d4bb-4bce-bfd8-290ae3662f18","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Expanding Triple Brackets","topicId":"c555ec2f-9d2d-4d36-8d53-7c5ebd9de8c2","topicSidebar":"$L3b","topicSlug":"grade-6-6-4-expanding-triple-brackets","topicTitle":"Expanding Triple Brackets"}]}],"$L3c","$L3d"]}]