Divide $\\displaystyle{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ by $\\simplify[std]{{r}x+{s}}$ so that:
\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=q(x)+\\frac{r}{\\simplify[std]{{r}x+{s}}}\\]
where $q(x)$ is the quotient polynomial and $r$ is the remainder ($r$ is a constant).
\nThe coefficients of $q(x)$ are integers, do not input as decimals.
\n ", "name": "Luis's copy of Divide Polynomials", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
"}, "ungrouped_variables": ["be", "s2", "s1", "m", "al", "n", "s", "r", "t"], "parts": [{"prompt": "\n \n \n$q(x)=\\;\\;$[[0]]
\n \n \n \nInput all numbers as integers and not as decimals.
\n \n \n \n$r=\\;\\;$[[1]]
\n \n \n ", "type": "gapfill", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "checkingaccuracy": 0.001, "variableReplacements": [], "showCorrectAnswer": true, "answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "vsetrange": [0, 1], "answersimplification": "std", "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "vsetrangepoints": 5, "type": "jme", "marks": 1, "checkingtype": "absdiff", "scripts": {}, "notallowed": {"partialCredit": 0, "message": "Input numbers as integers not decimals.
", "showStrings": false, "strings": ["."]}, "showpreview": true}, {"showPrecisionHint": false, "type": "numberentry", "allowFractions": false, "variableReplacements": [], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "minValue": "{t*n+be-t*s}", "maxValue": "{t*n+be-t*s}", "correctAnswerFraction": false, "scripts": {}, "marks": 2}], "variableReplacementStrategy": "originalfirst", "scripts": {}}], "variables": {"s1": {"description": "", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)"}, "r": {"description": "", "name": "r", "group": "Ungrouped variables", "templateType": "anything", "definition": "1"}, "be": {"description": "", "name": "be", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)"}, "s2": {"description": "", "name": "s2", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)"}, "t": {"description": "", "name": "t", "group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(-9..9)"}, "m": {"description": "", "name": "m", "group": "Ungrouped variables", "templateType": "anything", "definition": "1"}, "s": {"description": "", "name": "s", "group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..9)"}, "al": {"description": "", "name": "al", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)"}, "n": {"description": "", "name": "n", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)"}}, "extensions": [], "preamble": {"css": "", "js": ""}, "type": "question", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "advice": "We have:
\n\\[\\begin{eqnarray*} \\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}&=&\\simplify[std]{(x+{s})x^2+{n}x^2+{n*s+t}x+{t*n+be}}\\\\&=&\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+{t}x+{t*n+be}}\\\\ &=&\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+(x+{s})*{t}+{t*n+be-s*t}}\\\\ &=&\\simplify[std]{(x+{s})(x^2+{n}x+{t})+{t*n+be-s*t}} \\end{eqnarray*} \\]
\nHence
\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=\\simplify[std]{x^2+{n}x+{t}+{t*n+be-s*t}/({r}x+{s})}\\]