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{"name": "Partial Fractions: Proper Fractions 4b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Partial Fractions: Proper Fractions 4b", "tags": [], "metadata": {"description": "<p>Rewrite the expression $\\frac{cx+d}{kx^2+mx+n}$ as partial fractions in the form $\\frac{A}{kx+a}+\\frac{B}{x+b}$, where the quadratic $kx^2+mx+n=(kx+a)(x+b)$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as partial fractions:</p>\n<p>\\[ \\simplify{({c}x+{d})/({k}x^2+{a+k*b}x+{a*b})} .\\]</p>", "advice": "<p>To express\\[\\simplify{({c}x+{d})/({k}x^2+{a+k*b}x+{a*b})} \\] as partial fractions, we must first factorise the denominator into linear factors:</p>\n<p>\\[\\simplify{({c}x+{d})/({k}x^2+{a+k*b}x+{a*b})}=\\simplify{({c}x+{d})/(({k}x+{a})(x+{b}))}.\\]</p>\n<p>Now the denominator is factorised, we want to set this equal to the sum of 2 fractions with denominators $\\simplify{{k}x+{a}}$ and $\\simplify{x+{b}}$. Since these are both distinct linear factors, this tells us that the numerators will be constants, which we will call $A$ and $B$:</p>\n<p>\\[ \\simplify{({c}x+{d})/(({k}x+{a})(x+{b}))} = \\simplify{A/({k}x+{a}) + B/(x+{b})}.\\]</p>\n<p>To find the values of $A$ and $B$, we want to multiply this equation by the denominator of the left-hand side. This gives</p>\n<p>\\[ \\simplify{{c}x+{d}=A(x+{b})+B({k}x+{a})}.\\]</p>\n<p>There are 2 methods of finding $A$ and $B$. The first is to choose suitable values for $x$ which will eliminate one of the terms, and the other is to compare the coefficients of each side of the equation. We will cover both methods here.</p>\n<p><strong>Method 1:</strong></p>\n<p>To find $A$, we can eliminate $B$ by setting $\\simplify[fractionNumbers]{x={-a/k}}$:</p>\n<p>\\[ \\simplify[fractionNumbers]{{d-c*a/k}=A{b-a/k}} \\implies \\simplify[fractionNumbers]{A={(d-c*a/k)/(b-a/k)}}.\\]</p>\n<p>Similarly, to find B, we can eliminate $A$ by setting $\\simplify{x={-b}}$:</p>\n<p>\\[ \\simplify{{d-c*b}=B{a-k*b}} \\implies \\simplify[fractionNumbers]{B={(d-c*b)/(a-k*b)}}.\\]</p>\n<p>Therefore,&nbsp;</p>\n<p>{check}</p>\n<p><strong>Method 2:</strong></p>\n<p>By comparing the coefficients of the $x$-terms and the constant terms we can form a pair of simultaneous equations to find $A$ and $B$.</p>\n<p>\\[ \\begin{split} \\simplify{{c}x+{d}} &amp;\\,= \\simplify{A(x+{b})+B({k}x+{a})} \\\\ &amp;\\,= \\simplify{A*x+{b}A+{k}*B*x +{a}B} \\\\ &amp;\\,=\\simplify{(A+{k}B)x +{b}A+{a}B} . \\end{split} \\]</p>\n<p>\\[ \\begin{split}&amp;(x):\\quad \\var{c} &amp;\\,= \\simplify{A+{k}B} \\\\ &amp;(c):\\quad \\var{d} &amp;\\,= \\simplify{{b}A+{a}B} .\\end{split} \\]</p>\n<p>Hence,</p>\n<p>\\[A=\\simplify[fractionNumbers]{{Asol}},\\,B=\\simplify[fractionNumbers]{{Bsol}}, \\]</p>\n<p>and</p>\n<p>{check}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "(d-c*a/k)/(b-a/k)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9 except b*k)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": 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