// Numbas version: exam_results_page_options {"name": "Partial fraction breakdown A,B & C: simple linear factors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "parts": [{"prompt": "
If the partial fraction breakdown is given by:
\n\\(Q(s) =\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{s+\\var{c1}}\\)
\nCalculate the values of \\(A, B\\) and \\(C\\) and give your answers as fractions
\n\\(A=\\) [[0]]
\n\\(B=\\) [[1]]
\n\\(C=\\) [[2]]
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\n\\(Q(s)=\\frac{\\var{F}s^2+\\var{G}s+\\var{H}}{(S+\\var{a1})(s+\\var{b1})(s+\\var{c1})}\\)
", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "advice": "\\(\\frac{\\var{F}s^2+\\var{G}s+\\var{H}}{(S+\\var{a1})(s+\\var{b1})(s+\\var{c1})}=\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{s+\\var{c1}}\\)
\nMultiply across by \\((s+\\var{a1})(s+\\var{b1})(s+\\var{c1})\\)
\n\\(\\var{F}s^2+\\var{G}s+\\var{H}=A(s+\\var{b1})(s+\\var{c1})+B(s+\\var{a1})(s+\\var{c1})+C(s+\\var{a1})(s+\\var{b1})\\)
\nlet \\(s=-\\var{a1}\\)
\n\\(\\var{F}(-\\var{a1})^2+\\var{G}(-\\var{a1})+\\var{H}=A(\\simplify{{b1}-{a1}})(\\simplify{{c1}-{a1}})+B(0)+C(0)\\)
\n\\(\\simplify{{F}*{a1}^2-{G}*{a1}+{H}}=\\simplify{({b1}-{a1})*({c1}-{a1})}A\\)
\n\\(A=\\frac{\\simplify{{F}*{a1}^2-{G}*{a1}+{H}}}{\\simplify{({b1}-{a1})*({c1}-{a1})}}\\)
\nlet \\(s=-\\var{b1}\\)
\n\\(\\var{F}(-\\var{b1})^2+\\var{G}(-\\var{b1})+\\var{H}=A(0)+B(\\simplify{-{b1}+{a1}})(\\simplify{{c1}-{b1}})+C(0)\\)
\n\\(\\simplify{{F}*{b1}^2-{G}*{b1}+{H}}=\\simplify{(-{b1}+{a1})*({c1}-{b1})}B\\)
\n\\(B=\\frac{\\simplify{{F}*{b1}^2-{G}*{b1}+{H}}}{\\simplify{(-{b1}+{a1})*({c1}-{b1})}}\\)
\nlet \\(s=-\\var{c1}\\)
\n\\(\\var{F}(-\\var{c1})^2+\\var{G}(-\\var{c1})+\\var{H}=A(0)+B(0)+C(\\simplify{-{c1}+{a1}})(\\simplify{-{c1}+{b1}})\\)
\n\\(\\simplify{{F}*{c1}^2-{G}*{c1}+{H}}=\\simplify{(-{c1}+{a1})*(-{c1}+{b1})}C\\)
\n\\(C=\\frac{\\simplify{{F}*{c1}^2-{G}*{c1}+{H}}}{\\simplify{(-{c1}+{a1})*(-{c1}+{b1})}}\\)
", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}