// Numbas version: finer_feedback_settings
{"name": "Dividing polynomials 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Dividing polynomials 3", "tags": ["algebra", "compare coefficients", "comparing coefficients", "degree of a polynomial", "division of polynomials", "long division of polynomials", "polynomial division", "polynomials", "quotient polynomial", "remainder polynomial"], "advice": "\n    \n    \n    <p>You can use the method of long division. Here we show an alternative method.</p>\n    \n    \n    \n    <p>The question tells us we can write:<br/>\\[\\frac{f(x)}{g(x)}=\\simplify[std]{(x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1})/(x^2+{e1})=q(x)+r(x)/(x^2+{e1})}\\;\\;\\;\\dots(1)\\]<br/>Since the degree of $f(x)$ is $4$ and the degree of $g(x)$ is $2$, the degree of $q(x)$ is $2$ and we can write $q(x)=Ax^2+Bx+C$ for constants $A,\\;B,\\;C$.</p>\n    \n    \n    \n    <p>The degree of $r(x)$ is less than $2$ hence $r(x)=Dx+E$ for constants $D,\\;E$.</p>\n    \n    \n    \n    <p>Multiplying both sides of (1) by $\\simplify[std]{x^2+{e1}}$ gives:<br/>\\[\\simplify[std]{x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}=(Ax^2+Bx+C)(x^2+{e1})+Dx}+E\\]<br/>We can then compare coefficients to determine the constants  $A,\\;B,\\;C,\\;D,\\;E$.</p>\n    \n    \n    \n    <h4>Terms in $x^4$</h4>\n    \n    \n    \n    <p>$A=1$</p>\n    \n    \n    \n    <h4>Terms in $x^3$</h4>\n    \n    \n    \n    <p>$B=\\var{a1}$</p>\n    \n    \n    \n    <h4>Terms in $x^2$</h4>\n    \n    \n    \n    <p>$\\simplify[std]{{e1}A+C = {b1}} \\Rightarrow C = \\var{b1-e1}$</p>\n    \n    \n    \n    <h4>Terms in $x$</h4>\n    \n    \n    \n    <p>$\\simplify[std]{{e1}B+D = {c1}} \\Rightarrow D = \\var{c1-a1*e1}$</p>\n    \n    \n    \n    <h4>Constant terms</h4>\n    \n    \n    \n    <p>$\\simplify[std]{{e1}C}+E= \\var{d1} \\Rightarrow E = \\var{d1-(b1-e1)*e1}$</p>\n    \n    \n    \n    <p>Therefore:<br/>\\[\\frac{f(x)}{g(x)}=\\simplify[std]{x ^ 2 + {a1} *x  + {b1 -e1}+({c1-a1*e1}*x+{d1-e1*(b1-e1)})/(x^2+{e1})}\\]</p>\n    \n    \n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>$q(x)=\\;\\;$[[0]]</p>\n        <p>$r(x)=\\;\\;$[[1]]</p>\n        <p>Input all numbers as integers and not as decimals in both questions.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "x ^ 2 + {a1} *x  + {b1 -e1}", "type": "jme"}, {"notallowed": {"message": "<p>Input all numbers as integers and not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{c1-a1*e1}*x+{d1-e1*(b1-e1)}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n  <p>Divide $\\displaystyle{f(x)=\\simplify[std]{ x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}}}$ by $g(x)=\\simplify[std]{x^2+{e1}}$ so that:<br/>\\[\\frac{f(x)}{g(x)}=q(x)+\\frac{r(x)}{g(x)}\\]</p>\n    <p>where $q(x)$ is the quotient polynomial and $r(x)$ and the degree of $r(x)$ is less than the degree of $g(x)$.</p>\n    <p>The coefficients of $q(x)$ are integers, do not input as decimals.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a1": {"definition": "random(-5..5)", "name": "a1"}, "s": {"definition": "random(1,-1)", "name": "s"}, "b1": {"definition": "random(-5..5)", "name": "b1"}, "c1": {"definition": "random(-5..5)", "name": "c1"}, "e1": {"definition": "s*random(1..4)", "name": "e1"}, "d1": {"definition": "random(-5..5)", "name": "d1"}}, "metadata": {"notes": "\n                    \t\t<p><strong>29/06/2012:</strong></p>\n                    \t\t<p><strong><br /></strong>Added and edited tags.</p>\n                    \t\t<p>Added forbidden string, the decimal point, as it has been missed out.</p>\n                    \t\t<p><strong>19/07/2012:</strong></p>\n                    \t\t<p>Added description.</p>\n                    \t\t<p>Note that the solution given in Advice uses the comparison of coefficients method. Could also include two other methods - but no, too unwieldy!</p>\n                    \t\t<p>Checked calculations.</p>\n                    \t\t<p><strong>23/07/2012:</strong></p>\n                    \t\t<p>Added tags.</p>\n                    \t\t<p>Headings in the Advice section look too big. Consider writing in bold font instead of bigger font?</p>\n                    \t\t<p>Question appears to be working correctly.</p>\n                    \t\t<p>&nbsp;</p>\n                    \t\t", "description": "<p>Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:<br />$\\displaystyle \\frac{f(x)}{g(x)}=q(x)+\\frac{r(x)}{g(x)}$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}