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{"name": "Dividing a polynomial with remainders, using the remainder theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question tests the student's ability to find remainders using the remainder theorem.&nbsp;</p>"}, "tags": ["taxonomy"], "functions": {}, "name": "Dividing a polynomial with remainders, using the remainder theorem", "ungrouped_variables": ["coef_x3", "coef_x2", "coef_x", "const", "k", "remainder", "a"], "parts": [{"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": true, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "remainder", "scripts": {}, "marks": "2", "allowFractions": true, "type": "numberentry", "prompt": "<p>Find the remainder when $f(x) = \\simplify{{coef_x3}x^3+{coef_x2}x^2+{coef_x}x+{const}}$ is divided by $(\\simplify{{a}x+{k}})$, using the remainder theorem.</p>", "correctAnswerFraction": true, "maxValue": "remainder"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"coef_x2": {"description": "<p>Coefficient of x^2.</p>", "definition": "random(-3..3 except 0)", "name": "coef_x2", "templateType": "anything", "group": "Ungrouped variables"}, "remainder": {"description": "<p>Correct remainder.</p>", "definition": "coef_x3*(-k/a)^3+coef_x2*(-k/a)^2+coef_x*(-k/a)+const", "name": "remainder", "templateType": "anything", "group": "Ungrouped variables"}, "k": {"description": "<p>Free&nbsp;coefficient in the dividing equation.</p>", "definition": "random(-3..3 except 0 except 1 except -1)", "name": "k", "templateType": "anything", "group": "Ungrouped variables"}, "const": {"description": "<p>Constant term</p>", "definition": "random(-5..5 except 0)", "name": "const", "templateType": "anything", "group": "Ungrouped variables"}, "coef_x": {"description": "<p>Coefficient of x.</p>", "definition": "random(-3..3 except 0)", "name": "coef_x", "templateType": "anything", "group": "Ungrouped variables"}, "coef_x3": {"description": "<p>Coefficient of x^3</p>", "definition": "random(2..4)  ", "name": "coef_x3", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"description": "<p>Leading coefficient in the dividing equation.</p>", "definition": "random(1..3 except k except -k)", "name": "a", "templateType": "anything", "group": "Ungrouped variables"}}, "extensions": [], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "<p>The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{a*x-b})$ then the remainder is $f \\left( \\frac{b}{a} \\right)$.</p>\n<p>This means that if we substitute $x = \\frac{b}{a}$ into the equation for $f(x)$, the result will be equal to the remainder when $f(x)$ is divided by $(\\simplify{a*x-b})$.</p>\n<p>Therefore, to calculate the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$, we use this same principle.</p>\n<p>As we are dividing $f(x)$ by $(\\simplify{{a}*x+{k}})$, using the remainder theorem tells us that substituting</p>\n<p>\\[<br/>\\begin{align}<br/>x &amp;= \\frac{b}{a}\\\\<br/>&amp;= \\simplify{-({k}/{a})}&nbsp;<br/>\\end{align}<br/>\\]</p>\n<p>into our equation for $f(x)$ will give us the remainder when $f(x)$ is divided by $(\\simplify{{a}*x+{k}})$. Substituting this value of $x$ into $f(x)$ gives us</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-({k}/{a})}) &amp;= \\simplify[all,!collectNumbers, fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}\\\\<br/>&amp;= \\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}.<br/>\\end{align}<br/>\\]&nbsp;</p>\n<p>Therefore, the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$ is &nbsp;$\\displaystyle\\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}$.</p>", "statement": "", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}