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{"name": "Use the factor theorem to identify factors of a polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "b", "c", "d", "coef1_x3", "coef1_x2", "coef1_x", "const", "coef2_x3", "coef2_x2", "coef2_x", "coef3_x3", "coef3_x2", "coef3_x"], "tags": ["factor theorem", "Factor Theorem", "factors", "Factors", "Multiple choice", "Multiple Choice", "multiple choice", "polynomial", "Polynomial", "taxonomy"], "variable_groups": [], "functions": {}, "parts": [{"displayColumns": 0, "displayType": "checkbox", "matrix": ["1", 0, "1"], "warningType": "none", "type": "m_n_2", "maxMarks": 0, "minAnswers": 0, "showCorrectAnswer": true, "marks": 0, "scripts": {}, "maxAnswers": 0, "choices": ["<p>$(\\simplify{x+{a}})$</p>", "<p>$(\\simplify{x+{d}})$</p>", "<p>$(\\simplify{x+{c}})$</p>"], "prompt": "<p>Use the factor theorem to find&nbsp;which <strong>two</strong> of the following are&nbsp;factors of the&nbsp;polynomial</p>\n<p>\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]</p>", "showFeedbackIcon": true, "shuffleChoices": true, "variableReplacements": [], "minMarks": 0, "variableReplacementStrategy": "originalfirst", "distractors": ["", "", ""]}], "name": "Use the factor theorem to identify factors of a polynomial", "variables": {"coef1_x": {"description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "group": "Ungrouped variables", "name": "coef1_x", "templateType": "anything", "definition": "(a*b+b*c+a*c)*(-a)"}, "b": {"description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "group": "Ungrouped variables", "name": "b", "templateType": "anything", "definition": "random(-2..3 except 0 except c)"}, "d": {"description": "<p>Incorrect answer for part a.</p>", "group": "Ungrouped variables", "name": "d", "templateType": "anything", "definition": "random(-2..2 except 0 except a except c except b)"}, "coef1_x3": {"description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "group": "Ungrouped variables", "name": "coef1_x3", "templateType": "anything", "definition": "(-a)^3"}, "coef1_x2": {"description": "<p>Number obtained from putting x=-a into the second term of the equation.</p>", "group": "Ungrouped variables", "name": "coef1_x2", "templateType": "anything", "definition": "(a+b+c)*(-a)^2"}, "coef2_x3": {"description": "<p>Number obtained from putting x=-d into the&nbsp;first term&nbsp;in the&nbsp;equation.</p>", "group": "Ungrouped variables", "name": "coef2_x3", "templateType": "anything", "definition": "(-d)^3"}, "coef3_x": {"description": "<p>Number obtained by putting x=-c into the third term of the equation.</p>", "group": "Ungrouped variables", "name": "coef3_x", "templateType": "anything", "definition": "(a*b+b*c+a*c)*(-c)"}, "const": {"description": "<p>Constant term in the equation.</p>", "group": "Ungrouped variables", "name": "const", "templateType": "anything", "definition": "a*b*c"}, "coef2_x2": {"description": "<p>Number obtained from putting x=-d into the second term of the equation.</p>", "group": "Ungrouped variables", "name": "coef2_x2", "templateType": "anything", "definition": "(a+b+c)*(-d)^2"}, "coef2_x": {"description": "<p>Number obtained from putting x=-d into the 3rd term for the equation.</p>", "group": "Ungrouped variables", "name": "coef2_x", "templateType": "anything", "definition": "(a*b+b*c+a*c)*(-d)"}, "c": {"description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "group": "Ungrouped variables", "name": "c", "templateType": "anything", "definition": "random(-2..3 except 0)"}, "coef3_x3": {"description": "<p>Number obtained for putting x=-c into the first term of the equation.</p>", "group": "Ungrouped variables", "name": "coef3_x3", "templateType": "anything", "definition": "(-c)^3"}, "a": {"description": "<p>Random number between -2 and 3, not including 0 for creating polynomial.</p>", "group": "Ungrouped variables", "name": "a", "templateType": "anything", "definition": "random(-2..3 except 0 except c)"}, "coef3_x2": {"description": "", "group": "Ungrouped variables", "name": "coef3_x2", "templateType": "anything", "definition": "(a+b+c)*(-c)^2"}}, "rulesets": {}, "advice": "<p>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.&nbsp;</p>\n<p>If $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.</p>\n<p>If $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by&nbsp;the factor theorem,&nbsp;$f(\\simplify{-{a}}) = 0$.</p>\n<p>We see that</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{a}})&nbsp;&amp;= \\simplify[all,!collectNumbers]{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{a})})$ is a factor of $f(x)$.</p>\n<p>Similarly for $(\\simplify{(x+{d})})$,</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{d}})&nbsp;&amp;= \\simplify[all,!collectNumbers]{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\<br/>&amp;\\neq 0.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{d})})$ is not a factor of $f(x)$.</p>\n<p>Finally, for $(\\simplify{(x+{c})})$,</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{c}}) &amp;=&nbsp;\\simplify[all,!collectNumbers]{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{c})})$ is also a factor of $f(x)$.</p>", "statement": "<p>The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.&nbsp;</p>", "metadata": {"description": "<p>Apply the&nbsp;factor theorem to check which of a list of linear polynomials are factors of another polynomial.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "type": "question", "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}