Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
"}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input numbers as integers not decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{t*n+be-t*s}", "maxValue": "{t*n+be-t*s}", "variableReplacements": [], "marks": 2}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "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 ", "variableReplacements": [], "marks": 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 ", "tags": ["algebra", "algebraic manipulation", "checked2015", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "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})}\\]
Apply the factor theorem to check which of a list of linear polynomials are factors of another polynomial.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "To find the factors of the polynomial $f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}$, we use the factor theorem.
\nIf $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.
\nIf $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by the factor theorem, $f(\\simplify{-{a}}) = 0$.
\nWe see that
\n\\[
\\begin{align}
f(\\simplify{-{a}}) &= \\simplify[all,!collectNumbers]{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}\\\\
&= \\simplify{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}.
\\end{align}
\\]
Therefore, $(\\simplify{(x+{a})})$ is a factor of $f(x)$.
\nSimilarly for $(\\simplify{(x+{d})})$,
\n\\[
\\begin{align}
f(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers]{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\
&= \\simplify{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\
&\\neq 0.
\\end{align}
\\]
Therefore, $(\\simplify{(x+{d})})$ is not a factor of $f(x)$.
\nFinally, for $(\\simplify{(x+{c})})$,
\n\\[
\\begin{align}
f(\\simplify{-{c}}) &= \\simplify[all,!collectNumbers]{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}\\\\
&= \\simplify{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}.
\\end{align}
\\]
Therefore, $(\\simplify{(x+{c})})$ is also a factor of $f(x)$.
", "statement": "The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.
", "preamble": {"js": "", "css": ""}, "variables": {"coef1_x3": {"templateType": "anything", "description": "Number obtained from putting x=-a into the first term of the equation.
", "name": "coef1_x3", "group": "Ungrouped variables", "definition": "(-a)^3"}, "coef1_x2": {"templateType": "anything", "description": "Number obtained from putting x=-a into the second term of the equation.
", "name": "coef1_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-a)^2"}, "coef2_x3": {"templateType": "anything", "description": "Number obtained from putting x=-d into the first term in the equation.
", "name": "coef2_x3", "group": "Ungrouped variables", "definition": "(-d)^3"}, "coef1_x": {"templateType": "anything", "description": "Number obtained from putting x=-a into the first term of the equation.
", "name": "coef1_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-a)"}, "coef3_x3": {"templateType": "anything", "description": "Number obtained for putting x=-c into the first term of the equation.
", "name": "coef3_x3", "group": "Ungrouped variables", "definition": "(-c)^3"}, "const": {"templateType": "anything", "description": "Constant term in the equation.
", "name": "const", "group": "Ungrouped variables", "definition": "a*b*c"}, "c": {"templateType": "anything", "description": "Random number between -2 and 3 except 0 for creating polynomial.
", "name": "c", "group": "Ungrouped variables", "definition": "random(-2..3 except 0)"}, "d": {"templateType": "anything", "description": "Incorrect answer for part a.
", "name": "d", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except a except c except b)"}, "coef2_x": {"templateType": "anything", "description": "Number obtained from putting x=-d into the 3rd term for the equation.
", "name": "coef2_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-d)"}, "a": {"templateType": "anything", "description": "Random number between -2 and 3, not including 0 for creating polynomial.
", "name": "a", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef2_x2": {"templateType": "anything", "description": "Number obtained from putting x=-d into the second term of the equation.
", "name": "coef2_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-d)^2"}, "b": {"templateType": "anything", "description": "Random number between -2 and 3 except 0 for creating polynomial.
", "name": "b", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef3_x": {"templateType": "anything", "description": "Number obtained by putting x=-c into the third term of the equation.
", "name": "coef3_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-c)"}, "coef3_x2": {"templateType": "anything", "description": "", "name": "coef3_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-c)^2"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "choices": ["$(\\simplify{x+{a}})$
", "$(\\simplify{x+{d}})$
", "$(\\simplify{x+{c}})$
"], "matrix": ["1", 0, "1"], "displayColumns": 0, "distractors": ["", "", ""], "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "marks": 0, "maxAnswers": 0, "prompt": "Use the factor theorem to find which two of the following are factors of the polynomial
\n\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]
", "variableReplacements": [], "warningType": "none", "showFeedbackIcon": true, "displayType": "checkbox", "shuffleChoices": true, "showCorrectAnswer": true, "minAnswers": 0, "scripts": {}}], "tags": ["factor theorem", "Factor Theorem", "factors", "Factors", "Multiple choice", "Multiple Choice", "multiple choice", "polynomial", "Polynomial", "taxonomy"], "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "When typing answers, use the ^ key (SHIFT 6) to express powers.
\nE.g. Type x^3 to get $x^3$
", "feedbackmessages": []}, "contributors": [{"name": "Brad McAuliffe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4645/"}], "extensions": [], "custom_part_types": [], "resources": []}