// Numbas version: exam_results_page_options {"name": "JD's copy of Finding the missing value of a constant in a polynomial, using the Factor Theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "
The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.
", "extensions": [], "ungrouped_variables": ["w", "a", "b", "d", "coef_x3", "coef_x2", "coef_x"], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "variables": {"coef_x2": {"description": "Number obtained by putting x=-d into the second term of the equation.
", "definition": "(w*d+a+w*b)*(-d)^2", "templateType": "anything", "group": "Ungrouped variables", "name": "coef_x2"}, "d": {"description": "Used in creation of the polynomial.
", "definition": "random(-2..2 except 0 except a except b)", "templateType": "anything", "group": "Ungrouped variables", "name": "d"}, "a": {"description": "Random number between -2 and 3, not including 0 for creating polynomial.
", "definition": "random(-2..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a"}, "w": {"description": "Random number between 2,3,4.
", "definition": "random(2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "w"}, "coef_x": {"description": "Number obtained by putting x=-d into the third term of the equation.
", "definition": "(a*d+w*b*d+a*b)*(-d)", "templateType": "anything", "group": "Ungrouped variables", "name": "coef_x"}, "coef_x3": {"description": "Number obtained by putting x=-d into the first term of the equation.
", "definition": "(w)*(-d)^3", "templateType": "anything", "group": "Ungrouped variables", "name": "coef_x3"}, "b": {"description": "Random number between -2 and 3 except 0 for creating polynomial.
", "definition": "random(-2..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b"}}, "advice": "Using the factor theorem, we know that if $(x-a)$ is a factor of a polynomial $f(x)$, then $f(a)=0$.
\nWe are given that $(\\simplify{x+{d}})$ is a factor of $g(x) = \\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x+m}$.
\nBy the factor theorem, this means that $g(\\simplify{-{d}}) = 0$.
\nSubstituting $x=\\simplify{-{d}}$ into $g(x)$ gives
\n\\[
\\begin{align}
g(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers]{{coef_x3}+{coef_x2}+{coef_x}+m}\\\\
&=\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}.
\\end{align}
\\]
Therefore, as $g(\\simplify{-{d}}) = 0$, we have
\n\\[
\\begin{align}
\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}&=0\\\\
m&=\\simplify{-({coef_x3}+{coef_x2}+{coef_x})}.
\\end{align}
\\]
Given a factor of a cubic polynomial, factorise it fully by first dividing by the given factor, then factorising the remaining quadratic.
", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["Factor Theorem", "factor theorem", "polynomials", "Polynomials", "taxonomy"], "functions": {}, "parts": [{"prompt": "Given that $(\\simplify{x+{d}})$ is a factor of $g(x) = \\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x}+m$, find the value of $m$.
\n$m =$ [[0]].
\n", "variableReplacements": [], "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"variableReplacements": [], "marks": "2", "showpreview": true, "vsetrangepoints": 5, "scripts": {}, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1], "expectedvariablenames": [], "type": "jme", "answer": "{-({w}*({-d})^3+({w}*{d}+{a}+{w}*{b})*({-d})^2+({a}*{d}+{w}*{b}*{d}+{a}*{b})*{-d})}", "checkingaccuracy": 0.001, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}]}], "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}]}], "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}