Expand the expression $\\var{pmult}(\\var{pxcoeff}x+\\var{pconstant})$.
\n[[0]] $x$ + [[1]]
", "steps": [{"marks": 0, "variableReplacements": [], "prompt": "The number in front of the bracket is multiplying the bracketed term, that is, each term in the brackets.
\n\nFor example, $3(5x+6)$ means $3\\times (5x+6)$ which means $3\\times 5x+3\\times 6$, and so expanding $3(5x+6)$ gives $15x+18$.
", "type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}, {"gaps": [{"correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "{nmult*nxcoeff}", "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{nmult*nxcoeff}", "correctAnswerFraction": false, "showCorrectAnswer": true}, {"correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "{nmult*nconstant}", "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{nmult*nconstant}", "correctAnswerFraction": false, "showCorrectAnswer": true}], "stepsPenalty": "1", "marks": 0, "variableReplacements": [], "prompt": "Expand $\\var{nmult}(\\var{nxcoeff}a-\\var{-nconstant})$.
\n[[0]] $a$ + [[1]]
", "steps": [{"marks": 0, "variableReplacements": [], "prompt": "The number in front of the bracket is multiplying the bracketed term, that is, each term in the brackets. Further, recall that a negative multiplied by a negative is a positive.
\n\nFor example, $-3(5a-6)$ means $-3\\times (5a-6)$ which means $(-3)\\times 5a+(-3)\\times (-6)$, and so expanding $3(5a+6)$ gives $-15a+18$.
", "type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}, {"gaps": [{"correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "{-cx}", "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{-cx}", "correctAnswerFraction": false, "showCorrectAnswer": true}, {"correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "{-cy}", "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{-cy}", "correctAnswerFraction": false, "showCorrectAnswer": true}, {"correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "{-cc}", "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{-cc}", "correctAnswerFraction": false, "showCorrectAnswer": true}], "stepsPenalty": "1", "marks": 0, "variableReplacements": [], "prompt": "Expand $-(\\var{cx}x-\\var{-cy}y+\\var{cc})$.
\n[[0]] $x$ + [[1]] $y$ + [[2]]
", "steps": [{"marks": 0, "variableReplacements": [], "prompt": "A negative sign in front of a bracket is a common way to signify $-1$ times the bracketed term. The result is that it changes the sign of everything in the brackets.
\n\nFor example, $-(5x-y+6)$ means $-1\\times (5x-y+6)$ which means $(-1)\\times 5x+(-1)\\times (-y)+(-1)\\times 6$, and so expanding $-(5x-y+6)$ gives $-5x+y-6$.
", "type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}}], "variables": {"pmult": {"description": "", "group": "part a", "templateType": "anything", "definition": "primes[0]", "name": "pmult"}, "nxcoeff": {"description": "", "group": "part b", "templateType": "anything", "definition": "random(2..12)", "name": "nxcoeff"}, "pconstant": {"description": "", "group": "part a", "templateType": "anything", "definition": "primes[2]", "name": "pconstant"}, "nconstant": {"description": "", "group": "part b", "templateType": "anything", "definition": "random(-12..-2)", "name": "nconstant"}, "cc": {"description": "", "group": "part c", "templateType": "anything", "definition": "random(2..12)", "name": "cc"}, "nmult": {"description": "", "group": "part b", "templateType": "anything", "definition": "random(-12..-2)", "name": "nmult"}, "pxcoeff": {"description": "", "group": "part a", "templateType": "anything", "definition": "primes[1]", "name": "pxcoeff"}, "primes": {"description": "", "group": "part a", "templateType": "anything", "definition": "shuffle([2,3,5,7,11])[0..3]", "name": "primes"}, "cy": {"description": "", "group": "part c", "templateType": "anything", "definition": "random(-12..1)", "name": "cy"}, "cx": {"description": "", "group": "part c", "templateType": "anything", "definition": "random(2..12)", "name": "cx"}}, "functions": {}, "advice": "", "variable_groups": [{"variables": ["pmult", "pxcoeff", "pconstant", "primes"], "name": "part a"}, {"variables": ["nmult", "nxcoeff", "nconstant"], "name": "part b"}, {"variables": ["cx", "cy", "cc"], "name": "part c"}], "ungrouped_variables": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "rulesets": {}, "statement": "For all questions in this quiz, if you want to show indices (e.g., $x^2$) then input this as x^2 for $x^2$, y^3 for $y^3$, etc.
", "type": "question"}, {"name": "Expanding brackets", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}], "functions": {}, "ungrouped_variables": ["const1", "const2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "vsetrangepoints": 5, "prompt": "Expand and simplify the following algebraic expression:
\n$(x+\\var{const1})(x+\\var{const2})$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "You need to multiply everything in the first bracket by the second bracket. That is,
\n$x (x + \\var{const2}) + \\var{const1} (x + \\var{const2})=x^2+\\var{const2}x+\\var{const1}x+(\\var{const1}\\times\\var{const2})$
\nand then simplify your answer.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "answer": "x^2+({const1}+{const2})*x+({const1}*{const2})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "maxlength": {"length": "14", "message": "You need to simplify your solution as far as possible.
", "partialCredit": "50"}}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"const1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "const1", "description": ""}, "const2": {"definition": "random(2..9 except const1)", "templateType": "anything", "group": "Ungrouped variables", "name": "const2", "description": ""}}, "metadata": {"description": "This question tests the method of expanding a pair of brackets.
", "licence": "None specified"}, "type": "question"}, {"name": "Expansion of two brackets: Linear 2 positive coefficients", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of two linear terms", "Rebel", "REBEL", "rebel", "rebelmaths"], "advice": "\nUsing the method given by Show steps we have:
\n\\[\\begin{eqnarray*}\\simplify[std]{ ({a}x+{b})({c}x+{d})}&=&\\simplify[std]{{a}x*({c}x+{d})+{b}({c}x+{d})}\\\\&=&\\simplify[std]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}}\\\\&=&\\simplify[std]{{a*c}x^2+{(a*d+b*c)}x+{b*d}}\\end{eqnarray*}\\]
\n\n ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n
$\\simplify[std]{({a}x+{b})({c}x+{d})}=\\;$[[0]].
\nYour answer should be a quadratic in $x$ and should not include any brackets.
\nYou can click on Show steps for more information, but you will lose one mark if you do so.
\n ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.
", "showStrings": false, "strings": ["("], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "maxlength": {"length": 17, "message": "Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.
", "partialCredit": 0}, "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "musthave": {"message": "Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.
", "showStrings": false, "strings": ["x^2"], "partialCredit": 0}, "scripts": {}, "answer": "{a*c}x^2+{b*c+a*d}x+{b*d}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1], "answersimplification": "std"}], "steps": [{"prompt": "\nThere are many ways to expand an expression such as $(ax+b)(cx+d)$.
\nOne way:
\n\\[\\begin{eqnarray*} (ax+b)(cx+d)&=&ax(cx+d)+b(cx+d)\\\\&=&acx^2+adx+bcx+bd\\\\&=&acx^2+(ad+bc)x+bd\\end{eqnarray*}\\]
\n ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "Expand the following to give a quadratic in $x$.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(2..9 except [0,c])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "Expand $(ax+b)(cx+d)$.
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "feedback": {"showactualmark": true, "intro": "", "feedbackmessages": [], "allowrevealanswer": true, "advicethreshold": 0, "showtotalmark": true, "showanswerstate": true}, "percentPass": 0, "showQuestionGroupNames": false, "navigation": {"browse": true, "onleave": {"action": "none", "message": ""}, "reverse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true}, "showstudentname": true, "name": "Multiplying out brackets", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Questions about multiplying out brackets.
"}, "type": "exam", "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}], "extensions": [], "custom_part_types": [], "resources": []}