// Numbas version: exam_results_page_options {"name": "Marlon's copy of 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}, "tags": ["factor theorem", "Factor Theorem", "factors", "Factors", "Multiple choice", "Multiple Choice", "multiple choice", "polynomial", "Polynomial", "taxonomy"], "extensions": [], "parts": [{"showFeedbackIcon": true, "displayColumns": 0, "warningType": "none", "scripts": {}, "variableReplacements": [], "displayType": "checkbox", "showCorrectAnswer": true, "type": "m_n_2", "minMarks": 0, "distractors": ["", "", ""], "choices": ["

$(\\simplify{x+{a}})$

", "

$(\\simplify{x+{d}})$

", "

$(\\simplify{x+{c}})$

"], "shuffleChoices": true, "minAnswers": 0, "marks": 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}}.\\]

", "maxMarks": 0, "maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "matrix": ["1", 0, "1"]}], "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"], "preamble": {"css": "", "js": ""}, "variable_groups": [], "rulesets": {}, "name": "Marlon's copy of Use the factor theorem to identify factors of a polynomial", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Apply the factor theorem to check which of a list of linear polynomials are factors of another polynomial.

"}, "variables": {"coef1_x2": {"description": "

Number obtained from putting x=-a into the second term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b+c)*(-a)^2", "name": "coef1_x2"}, "coef3_x2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b+c)*(-c)^2", "name": "coef3_x2"}, "coef1_x": {"description": "

Number obtained from putting x=-a into the first term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-a)", "name": "coef1_x"}, "coef2_x2": {"description": "

Number obtained from putting x=-d into the second term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b+c)*(-d)^2", "name": "coef2_x2"}, "b": {"description": "

Random number between -2 and 3 except 0 for creating polynomial.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)", "name": "b"}, "coef3_x": {"description": "

Number obtained by putting x=-c into the third term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-c)", "name": "coef3_x"}, "coef3_x3": {"description": "

Number obtained for putting x=-c into the first term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(-c)^3", "name": "coef3_x3"}, "a": {"description": "

Random number between -2 and 3, not including 0 for creating polynomial.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)", "name": "a"}, "d": {"description": "

Incorrect answer for part a.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except a except c except b)", "name": "d"}, "coef1_x3": {"description": "

Number obtained from putting x=-a into the first term of the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(-a)^3", "name": "coef1_x3"}, "const": {"description": "

Constant term in the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "a*b*c", "name": "const"}, "c": {"description": "

Random number between -2 and 3 except 0 for creating polynomial.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..3 except 0)", "name": "c"}, "coef2_x": {"description": "

Number obtained from putting x=-d into the 3rd term for the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-d)", "name": "coef2_x"}, "coef2_x3": {"description": "

Number obtained from putting x=-d into the first term in the equation.

", "templateType": "anything", "group": "Ungrouped variables", "definition": "(-d)^3", "name": "coef2_x3"}}, "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. 

\n

If $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

\n

If $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by the factor theorem, $f(\\simplify{-{a}}) = 0$.

\n

We 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}
\\]

\n

Therefore, $(\\simplify{(x+{a})})$ is a factor of $f(x)$.

\n

Similarly 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}
\\]

\n

Therefore, $(\\simplify{(x+{d})})$ is not a factor of $f(x)$.

\n

Finally, 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}
\\]

\n

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 $f(x)$. 

", "type": "question", "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}]}], "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}