// Numbas version: exam_results_page_options {"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": [{"extensions": [], "rulesets": {}, "name": "Use the factor theorem to identify factors of a polynomial", "functions": {}, "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"], "metadata": {"description": "
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.
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", "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": ["$(\\simplify{x+{a}})$
", "$(\\simplify{x+{d}})$
", "$(\\simplify{x+{c}})$
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\n\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]
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