// Numbas version: exam_results_page_options {"name": "Amy'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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variables": {"coef1_x2": {"description": "

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

", "definition": "(a+b+c)*(-a)^2", "templateType": "anything", "name": "coef1_x2", "group": "Ungrouped variables"}, "coef1_x3": {"description": "

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

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

Constant term in the equation.

", "definition": "a*b*c", "templateType": "anything", "name": "const", "group": "Ungrouped variables"}, "coef1_x": {"description": "

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

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

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

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

", "definition": "random(-2..3 except 0 except c)", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "coef3_x3": {"description": "

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

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

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

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

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

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

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

", "definition": "(a*b+b*c+a*c)*(-c)", "templateType": "anything", "name": "coef3_x", "group": "Ungrouped variables"}, "coef2_x2": {"description": "

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

", "definition": "(a+b+c)*(-d)^2", "templateType": "anything", "name": "coef2_x2", "group": "Ungrouped variables"}, "coef2_x3": {"description": "

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

", "definition": "(-d)^3", "templateType": "anything", "name": "coef2_x3", "group": "Ungrouped variables"}, "coef3_x2": {"description": "", "definition": "(a+b+c)*(-c)^2", "templateType": "anything", "name": "coef3_x2", "group": "Ungrouped variables"}, "coef2_x": {"description": "

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

", "definition": "(a*b+b*c+a*c)*(-d)", "templateType": "anything", "name": "coef2_x", "group": "Ungrouped variables"}}, "name": "Amy's copy of Use the factor theorem to identify factors of a polynomial", "functions": {}, "tags": ["factor theorem", "Factor Theorem", "factors", "Factors", "Multiple choice", "Multiple Choice", "multiple choice", "polynomial", "Polynomial", "taxonomy"], "statement": "

The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.

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

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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 $f(x)$.

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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}}.\$

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

", "

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

", "

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

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