// Numbas version: exam_results_page_options {"name": "Expansion of two brackets: Linear and Quadratic ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Expansion of two brackets: Linear and Quadratic ", "tags": ["algebra", "algebraic manipulation", "expansion of a quadratic and linear term", "expansion of brackets"], "advice": "\n

Using the method given by Show steps:

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\\[\\begin{eqnarray*}\\simplify[std]{ ({p}w+{q})({a}w^2+{b}w+{c})}&=&\\simplify[std]{{p}w*({a}w^2+{b}w+{c})+{q}({a}w^2+{b}w+{c})}\\\\&=&\\simplify[std]{{a*p}w^3+{b*p}w^2+{c*p}w+{a*q}w^2+{q*b}w+{c*q}}\\\\&=&\\simplify[std]{{a*p}w^3+{(a*q+b*p)}w^2+{b*q+c*p}w+{c*q}}\\end{eqnarray*}\\]

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$\\simplify[std]{({p}w+{q})({a}w^2+{b}w+{c})}=\\;$[[0]].

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Your answer should be a cubic in $w$ and should not include any brackets.

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You can click on Show steps for more information, but you will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Do not include brackets in your answer. Input your answer as a cubic in $w$, in the form $aw^3+bw^2+cw+d$ for appropriate integers $a,\\;b,\\;c,\\;d$.

", "showstrings": false, "strings": ["(", "ww", "w*w"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{a*p}w^3+{a*q+p*b}w^2+{q*b+c*p}w+{c*q}", "type": "jme"}], "steps": [{"prompt": "\n

One way to expand this is as follows:

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$(pw+q)(aw^2+bw+c)=pw (aw^2+bw+c) +q(aw^2+bw+c)$ etc..

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Or as $(pw+q)(aw^2+bw+c)=(aw^2+bw+c)(pw+q)$ we can expand it as:

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$(aw^2+bw+c)(pw+q)=aw^2(pw+q)+bw(pw+q)+c(pw+q)$ 

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Expand the following to give a cubic in $w$.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "random(2..5)", "name": "c"}, "b": {"definition": "random(-9..9 except [0,a])", "name": "b"}, "d": {"definition": "random(-9..9 except [0,c])", "name": "d"}, "q": {"definition": "random(-3..3 except [0,b,d])", "name": "q"}, "p": {"definition": "random(1..3 except [a,c])", "name": "p"}}, "metadata": {"notes": "\n \t\t \t\t \t\t

16/08/2012:

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Added tags.

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Expand $(pw+q)(aw^2+bw+c)$.

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