This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.
", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "We are told that the polynomial:
\nFirstly, substituting $x = \\simplify{-{c}}$ into $p(x)$ gives us
\n\\begin{align}
p(\\simplify{-{c}}) &= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t},\\\\
&= \\simplify[all,fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t}.
\\end{align}
But, by the remainder theorem $p(\\simplify{-{c}}) = \\var{rem1}$ (using the first bullet point), so this becomes
\n\\begin{align}
\\simplify[all,fractionnumbers]{{coef2_x3*(-{c})^3}+s*{(-{c})^2}+{coef2_x*(-{c})}+t} &= \\var{rem1},\\\\
\\simplify[all,fractionnumbers]{s*{x}+t} &= \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}}.
\\end{align}
Similarly, substituting $x = \\simplify{-{d}}$ into $p(x)$, gives us
\n\\begin{align}
p(\\simplify{-{d}}) &= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t},\\\\
&= \\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t}.
\\end{align}
But, by the remainder theorem $p(\\simplify{-{d}}) = \\var{rem2}$ (using the second bullet point), so this becomes
\n\\begin{align}
\\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+s*{(-{d})^2}+{coef2_x*(-{d})}+t} &= \\var{rem2},\\\\
\\simplify[all,fractionnumbers]{s*{y}+t} &= \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.
\\end{align}
We now have two simultaneous equations for $s$ and $t$:
\n\\begin{align}
\\simplify[all,fractionnumbers]{s*{x}+t} = \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}} \\\\
\\simplify[all,fractionnumbers]{s*{y}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}
\\end{align}
Next, we subtract the second equation from the first equation.
\nThis allows us to cancel out the terms involving $t$ and gives us an equation only in terms of $s$, which we can then rearrange to find the value of $s$.
\nSubtracting the two equations gives
\n\\[\\simplify{s*{(-{c})^2-(-{d})^2}} = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}}.\\]
\nThen, we can rearrange this equation so that
\n\\[s = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{{(-c)^2-(-d)^2}}}.\\]
\nWe can calculate $t$ by substituting our value of $s$ into one of our original simultaneous equations. For example, let's use the equation
\n\\[\\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.\\]
\nSubstituting our value of $s$ into this equation gives us
\n\\[
\\begin{align}
\\simplify[all,fractionnumbers,!noleadingMinus]{{numerator/denominator}+t} &= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d)}},\\\\
t &= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d) - numerator/denominator}}.
\\end{align}
\\]
This same answer would've also been obtained if we had substituted our value of $s$ into the other equation instead.
", "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{c}^2", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "{c}^2", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{rem1}+{coef2_x3}*(c^3)+{coef2_x}*(c)", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "{rem1}+{coef2_x3}*(c^3)+{coef2_x}*(c)", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}], "sortAnswers": false, "showFeedbackIcon": true, "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "prompt": "Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{c}})$, create an equation involving $s$ and $t$.
\n[[0]]$s + t$ = [[1]].
\n", "type": "gapfill", "customMarkingAlgorithm": "", "steps": [{"showCorrectAnswer": true, "scripts": {}, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "prompt": "The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{x-b})$ then the remainder is $f(b)$.
", "type": "information", "variableReplacements": [], "marks": 0}], "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{d}^2", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "{d}^2", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{d}})$, create another equation involving $s$ and $t$.
\n[[0]]$s+t$ = [[1]].
\n", "type": "gapfill", "customMarkingAlgorithm": "", "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{(-c)^2-(-d)^2}", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{(-c)^2-(-d)^2}", "correctAnswerFraction": true, "customMarkingAlgorithm": "", "allowFractions": true, "correctAnswerStyle": "plain", "mustBeReduced": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "unitTests": []}], "sortAnswers": false, "showFeedbackIcon": true, "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "prompt": "Find the value of $s$. Reduce your answer to its simplest fractional form.
\n$s =$ [[0]]
", "type": "gapfill", "customMarkingAlgorithm": "", "steps": [{"showCorrectAnswer": true, "scripts": {}, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "prompt": "Subtract the two simultaneous equations for $s$ and $t$, obtained in parts a) and b), from each other.
\nThen rearrange this new equation to find the value of $s$.
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\n$t =$ [[0]]
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\nThen, rearrange this equation to find the value of $t$.
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", "templateType": "anything", "group": "Ungrouped variables"}, "x": {"name": "x", "definition": "(-(c))^2", "description": "Simplifies first coefficient of s.
", "templateType": "anything", "group": "Ungrouped variables"}, "denominator": {"name": "denominator", "definition": "(-c)^2-(-d)^2", "description": "Denominator of s.
", "templateType": "anything", "group": "Ungrouped variables"}, "numerator": {"name": "numerator", "definition": "(rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))*(-d)^2", "description": "Numerator of s
", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"name": "c", "definition": "random(-3..3 except 0 except 1)", "description": "Dividing term 1.
", "templateType": "anything", "group": "Ungrouped variables"}, "y": {"name": "y", "definition": "(-(d))^2", "description": "Simplifies second coefficient of s.
", "templateType": "anything", "group": "Ungrouped variables"}, "coef2_x3": {"name": "coef2_x3", "definition": "random(1..2)", "description": "Coefficient of x^3.
", "templateType": "anything", "group": "Ungrouped variables"}, "rem2": {"name": "rem2", "definition": "random(-3..3 except 0) ", "description": "Second remainder.
", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"name": "d", "definition": "random(-3..3 except c except -c except 0 except 1)", "description": "Dividing term 2.
", "templateType": "anything", "group": "Ungrouped variables"}, "coef2_x": {"name": "coef2_x", "definition": "random(-3..3 except 0 except c except d)", "description": "Coefficient of x.
", "templateType": "anything", "group": "Ungrouped variables"}}, "statement": "Consider the polynomial
\n\\[ p(x) = \\simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\\text{.}\\]
\nThe polynomial:
\n