// Numbas version: exam_results_page_options {"name": "Maths Support: Polynomial division", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "
2 questions. First question divides a cubic by a linear polynomial. The second divides a degree 4 polynomial by a degree 2 polynomial.
", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Dividing Polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "advice": "\nWe 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})}\\]
$q(x)=\\;\\;$[[0]]
\n \n \n \nInput all numbers as integers and not as decimals.
\n \n \n \n$r=\\;\\;$[[1]]
\n \n \n \n ", "gaps": [{"notallowed": {"message": "Input numbers as integers not decimals.
", "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": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "type": "jme"}, {"minvalue": "{t*n+be-t*s}", "type": "numberentry", "maxvalue": "{t*n+be-t*s}", "marks": 2.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "\nDivide $\\displaystyle{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ 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 \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"be": {"definition": "random(-9..9)", "name": "be"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": 1.0, "name": "m"}, "al": {"definition": "random(-9..9)", "name": "al"}, "n": {"definition": "random(-9..9)", "name": "n"}, "s": {"definition": "s1*random(1..9)", "name": "s"}, "r": {"definition": 1.0, "name": "r"}, "t": {"definition": "s2*random(-9..9)", "name": "t"}}, "metadata": {"notes": "\n \t\t \t\t28/6/2012:
\n \t\t \t\tChanged the divisor to $x+a$ where $a \\neq 0$, before this $a=0$ was allowed making the question easy.
\n \t\t \t\tAdded decimal point . as forbidden string to stop decimal input (is this necessary?)
\n \t\t \t\tAdded tags.
\n \t\t \t\tThe solution is given in terms of writing the dividend polynomial as powers of the linear divisor polynomial rather than using standard polynomial long division.
\n \t\t \t\t18/07/2012:
\n \t\t \t\tAdded description.
\n \t\t \t\t23/07/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \t\t\n \t\t \n \t\t", "description": "
Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Dividing polynomials 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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 \nYou can use the method of long division. Here we show an alternative method.
\n \n \n \nThe question tells us we can write:
\\[\\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)\\]
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 $A,\\;B,\\;C$.
The degree of $r(x)$ is less than $2$ hence $r(x)=Dx+E$ for constants $D,\\;E$.
\n \n \n \nMultiplying both sides of (1) by $\\simplify[std]{x^2+{e1}}$ gives:
\\[\\simplify[std]{x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}=(Ax^2+Bx+C)(x^2+{e1})+Dx}+E\\]
We can then compare coefficients to determine the constants $A,\\;B,\\;C,\\;D,\\;E$.
$A=1$
\n \n \n \n$B=\\var{a1}$
\n \n \n \n$\\simplify[std]{{e1}A+C = {b1}} \\Rightarrow C = \\var{b1-e1}$
\n \n \n \n$\\simplify[std]{{e1}B+D = {c1}} \\Rightarrow D = \\var{c1-a1*e1}$
\n \n \n \n$\\simplify[std]{{e1}C}+E= \\var{d1} \\Rightarrow E = \\var{d1-(b1-e1)*e1}$
\n \n \n \nTherefore:
\\[\\frac{f(x)}{g(x)}=\\simplify[std]{x ^ 2 + {a1} *x + {b1 -e1}+({c1-a1*e1}*x+{d1-e1*(b1-e1)})/(x^2+{e1})}\\]
$q(x)=\\;\\;$[[0]]
\n$r(x)=\\;\\;$[[1]]
\nInput all numbers as integers and not as decimals in both questions.
\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": "Input all numbers as integers and not as decimals.
", "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}], "statement": "\nDivide $\\displaystyle{f(x)=\\simplify[std]{ x ^ 4 + {a1} * x ^ 3 + {b1} * x^2 + {c1}x+{d1}}}$ by $g(x)=\\simplify[std]{x^2+{e1}}$ so that:
\\[\\frac{f(x)}{g(x)}=q(x)+\\frac{r(x)}{g(x)}\\]
where $q(x)$ is the quotient polynomial and $r(x)$ and the degree of $r(x)$ is less than the degree of $g(x)$.
\nThe coefficients of $q(x)$ are integers, do not input as decimals.
\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\t29/06/2012:
\n \t\t
Added and edited tags.
Added forbidden string, the decimal point, as it has been missed out.
\n \t\t19/07/2012:
\n \t\tAdded description.
\n \t\tNote that the solution given in Advice uses the comparison of coefficients method. Could also include two other methods - but no, too unwieldy!
\n \t\tChecked calculations.
\n \t\t23/07/2012:
\n \t\tAdded tags.
\n \t\tHeadings in the Advice section look too big. Consider writing in bold font instead of bigger font?
\n \t\tQuestion appears to be working correctly.
\n \t\t\n \t\t", "description": "
Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:
$\\displaystyle \\frac{f(x)}{g(x)}=q(x)+\\frac{r(x)}{g(x)}$