The polynomial {poly} has [[0]] (complex) roots (counted with multiplicity).
\nThat is, it can be written as the product of a constant (complex) factor and [[0]] linear factors $x-r_i$ where $r_i\\in \\mathbb{C}$ is a root of the polynomial.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{degree}", "minValue": "{degree}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "The Fundamental Theorem of Algebra can be stated as follows: every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ roots.
\n\nSince real numbers are complex numbers (just with an imaginary part of zero) and the highest power in the given polynomial is $\\var{degree}$, the Fundamental Theorem of Algebra can be applied and therefore the polynomial has exactly $\\var{degree}$ roots (counted with multiplicity). That is, there isn't necessarily $\\var{degree}$ unique roots but the polynomial can be written as the product of $\\var{degree}$ linear factors $x-r_i$ where $r_i\\in \\mathbb{C}$ is a root of the polynomial.
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\nThis guarantees us that $f\\large($ [[0]] $\\large)=$ [[1]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "The Remainder Theorem says that the remainder of dividing $f(x)$ by $x-a$ is equal to $f(a)$. So in this case we know $f(\\var{c})=\\var{rem}$.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{c}", "minValue": "{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{rem}", "minValue": "{rem}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "random(-12..12 except [0,n])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "n": {"definition": "random(3..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "rem": {"definition": "random(-12..12 except [c,n])", "templateType": "anything", "group": "Ungrouped variables", "name": "rem", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "The Factor Theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["Factors", "factors", "polynomial long division", "polynomials", "remainders", "roots"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "statement": "", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "distractors": ["", "", "", ""], "maxMarks": 0, "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "shuffleChoices": false, "matrix": ["1", 0, 0, 0], "showFeedbackIcon": true, "type": "1_n_2", "prompt": "Suppose a polynomial $f(x)$ evaluated at $\\var{c}$ is $0$, that is, $f(\\var{c})=0$.
\nWhich of the following is guaranteed?
", "steps": [{"showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "The Factor Theorem says that $x-a$ is a factor of a polynomial $f(x)$ if and only if $f(a)=0$.
\nSince we are told that $f(\\var{c})=0$ then we know $\\simplify{x-{c}}$ is a factor of $f(x)$.
", "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": []}], "displayType": "radiogroup", "marks": 0, "minMarks": 0, "choices": ["$\\simplify{x-{c}}$ is a factor of $f(x)$
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"], "displayColumns": 0}, {"showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "Suppose $\\simplify{x-{d}}$ is a factor of a polynomial $g(x)$. That is, if we divided $g(x)$ by $\\simplify{x-{d}}$ the remainder would be $0$.
\nThis guarantees us that $g\\large($ [[0]] $\\large)=$ [[1]].
", "steps": [{"showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "The Factor Theorem says that $x-a$ is a factor of a polynomial $f(x)$ if and only if $f(a)=0$.
\nSince we are told $\\simplify{x-{d}}$ is a factor of $g(x)$, we know that $g(\\var{d})=0$.
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", "matrix": ["1", 0, 0, 0], "shuffleChoices": false, "maxMarks": 0, "variableReplacements": [], "choices": ["constant.
", "linear function.
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"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "When you divide the dividend by the divisor, the remainder must be a polynomial of a smaller degree than the divisor.
\nIn the case that the divisor is $\\simplify{x-{c}}$ (a polynomial of degree 1) the remainder must be a polynomial of 0th degree, that is, a constant.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "distractors": ["", "", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "displayColumns": 0, "prompt": "Suppose you divide a polynomial of degree $\\var{n}$ by a polynomial of degree $\\var{m}$, then we can say the remainder is a polynomial of degree no more than
", "matrix": [0, "1", 0, 0], "shuffleChoices": true, "maxMarks": 0, "variableReplacements": [], "choices": ["$\\var{m}$
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\nIn the case that the divisor is a polynomial of degree $\\var{m}$ the remainder must be a polynomial of degree $\\var{m-1}$ or less.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "distractors": ["", "", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "displayColumns": 0, "prompt": "Again, suppose you divide a polynomial of degree $\\var{n}$ by a polynomial of degree $\\var{m}$, then we can say the quotient is a polynomial of degree
", "matrix": ["1", 0, 0, 0, 0], "shuffleChoices": true, "maxMarks": 0, "variableReplacements": [], "choices": ["$\\var{n-m}$
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\nThis means that the degree of the quotient must be $\\var{n}-\\var{m}=\\var{n-m}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "distractors": ["", "", "", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "m": {"definition": "random(3..n)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "n": {"definition": "random(5..14)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Divide Polynomials: Cubic by linear", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "advice": "There is a procedure called polynomial long division, if you are comfortable with long division of numbers then it isn't too different. You can see an example of the procedure here. However, if you don't recall long division we can rewrite the numerator so that it includes a multiple of the denominator. We do this term by term starting with the leading term (that is, $\\simplify{{r * m}x ^ 3}$).
\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})}\\]
Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
"}, "parts": [{"variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"variableReplacements": [], "checkvariablenames": false, "vsetrange": [0, 1], "showFeedbackIcon": true, "marks": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showpreview": true, "notallowed": {"showStrings": false, "strings": ["."], "message": "Input numbers as integers not decimals.
", "partialCredit": 0}, "type": "jme", "expectedvariablenames": [], "scripts": {}, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "std", "checkingaccuracy": 0.001, "answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})"}, {"variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true, "marks": 2, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain", "type": "numberentry", "minValue": "{t*n+be-t*s}", "correctAnswerFraction": false, "maxValue": "{t*n+be-t*s}"}], "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 "}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "tags": ["Algebra", "algebra", "algebraic manipulation", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "functions": {}, "ungrouped_variables": ["be", "s2", "s1", "m", "al", "n", "s", "r", "t"], "statement": "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.
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