rk M1)*"},{"type":"text","order":3,"title":"Simplify to get the final answer","content":"Finally, we collect all the terms from inside our second grid:\n$$x^3 + 7x^2 + 2x^2 + 12x + 14x + 24$$\n\nNow we combine the like terms:\n- Combine $x^2$ terms: $7x^2 + 2x^2 = 9x^2$\n- Combine $x$ terms: $12x + 14x = 26x$\n\nThis gives us our fully simplified cubic expression:\n$$x^3 + 9x^2 + 26x + 24$$\n\n*(This earns the final accuracy mark A1)*"}]},{"id":"b03f8322-05bc-41db-accc-85c67f6ab4aa:polynomial_identity","label":"Using Polynomial Identity","steps":[{"type":"text","order":0,"title":"Identify the identity","content":"To expand and simplify the expression $(x + 3)(x + 4)(x + 2)$, we can use the polynomial identity for the product of three binomials:\n\n$$(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$$","highlights":[{"text":"(x + 3)(x + 4)(x + 2)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a = \\htmlData{anchor=val-a}{\\color{@sky}{3}}, \\quad b = \\htmlData{anchor=val-b}{\\color{@amber}{4}}, \\quad c = \\htmlData{anchor=val-c}{\\color{@purple}{2}}$"},{"latex":"$$(x + \\htmlData{anchor=var-a}{\\color{@sky}{a}})(x + \\htmlData{anchor=var-b}{\\color{@amber}{b}})(x + \\htmlData{anchor=var-c}{\\color{@purple}{c}}) = x^3 + (\\htmlData{anchor=sum-a}{\\color{@sky}{a}} + \\htmlData{anchor=sum-b}{\\color{@amber}{b}} + \\htmlData{anchor=sum-c}{\\color{@purple}{c}})x^2 + (\\htmlData{anchor=prod-ab}{\\color{@sky}{a}}\\htmlData{anchor=prod-ba}{\\color{@amber}{b}} + \\htmlData{anchor=prod-bc}{\\color{@amber}{b}}\\htmlData{anchor=prod-cb}{\\color{@purple}{c}} + \\htmlData{anchor=prod-ca}{\\color{@purple}{c}}\\htmlData{anchor=prod-ac}{\\color{@sky}{a}})x + \\htmlData{anchor=term-abc}{\\color{@sky}{a}}\\htmlData{anchor=term-bca}{\\color{@amber}{b}}\\htmlData{anchor=term-cab}{\\color{@purple}{c}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"amber","style":"text-color","anchor":"val-b"},{"color":"purple","style":"text-color","anchor":"val-c"}],"connections":[{"to":"var-a","from":"val-a","color":"sky","label":"a = 3","style":"solid"},{"to":"var-b","from":"val-b","color":"amber","label":"b = 4","style":"solid"},{"to":"var-c","from":"val-c","color":"purple","label":"c = 2","style":"solid"}]},"order":1,"title":"Substitute the values","description":"We identify $a = 3$, $b = 4$, and $c = 2$ from our given expression $(x + 3)(x + 4)(x + 2)$ and substitute them into the identity."},{"type":"text","order":2,"title":"Calculate the coefficients","content":"Now, we compute each coefficient term:\n\n1. **Coefficient of $x^2$**:\n $$a + b + c = 3 + 4 + 2 = 9$$\n\n2. **Coefficient of $x$**:\n $$ab + bc + ca = (3 \\times 4) + (4 \\times 2) + (2 \\times 3) = 12 + 8 + 6 = 26$$\n\n3. **Constant term**:\n $$abc = 3 \\times 4 \\times 2 = 24$$"},{"type":"text","order":3,"title":"Write the final expression","content":"Plugging these values back into the polynomial identity, we get:\n\n$$x^3 + 9x^2 + 26x + 24$$\n\nThis is the fully expanded and simplified form of the expression."}]}],"generatedAt":"2026-06-13T03:50:50.164Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"cb807ffe-aab1-4210-9799-0e7e3ae64585","revisionId":"fc387abf-e2b8-409a-9182-716c29524a33","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"2a5cec5c-2d37-4e63-86db-d56c0ecaa2b9","label":"a","content":"Expand and Simplify $(x + 3)(x + 5)(x + 2)$","markscheme":"- Expands two brackets to get $x^2 + 8x + 15$ or $x^2 + 5x + 6$ or $x^2 + 7x + 10$ M1\n- Multiplies the resulting quadratic by the third linear bracket M1\n- Final simplified expression $x^3 + 10x^2 + 31x + 30$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Attempts to expand two of the brackets to obtain a quadratic expression with at least

Expanding Triple Brackets

Question 2

Expanding Triple Brackets Questions

Expanding Triple Brackets

Question 2

Expanding Triple Brackets Questions