// Numbas version: exam_results_page_options {"name": "Partial Fractions 3 - double root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "nb", "s1", "a_", "c_", "b_", "nb_", "a1_", "a2_", "s1_", "new", "a1", "a2", "a3", "d", "d_", "b1", "b2", "b3", "p", "q", "c1", "c2", "c3", "p1", "q1"], "name": "Partial Fractions 3 - double root", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "preamble": {"css": "", "js": ""}, "advice": "

a)

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We use partial fractions to find \$A\$, \$B\$ and \$C\$ such that:
\$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{A/(x+{a})+B/(x+{a})^2+C/(x+{b})}\$

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Dividing both sides of the equation by \$\\displaystyle \\simplify[std]{1/( (x+{a})^2(x+{b}) )}\\;\\;\$ we obtain:

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\$ \\simplify{A(x+{a})(x+{b})+B(x+{b})+C(x+{a})^2 = {a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}\$

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\$\\Rightarrow \\simplify[std]{(A+C)x^2+({a+b}A+B+{2a}C)x+({a*b}A+{b}B+{a*a}C)={a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}\$

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Identifying coefficients:

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Coefficient \$x^2\$: \$\\simplify[std]{A+C={a1+a3} }\$

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Coefficent \$x\$: \$ \\simplify[std]{ {a+b}A+B+{2a}C = {a1*a+a1*b+a2+2*a*a3} }\$

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Constant term: \$\\simplify{{a*b}A+{b}B+{a*a}C ={a1*a*b + a2*b + a3*a^2}}\$

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On solving these equations we obtain \$A = \\var{a1}\$, \$B=\\var{a2}\$ and \$C=\\var{a3}\$

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Which gives:\$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{{a1}/(x+{a})+{a2}/(x+{a})^2+{a3}/(x+{b})}\$

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Apply same method to solve b) and c)

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Split \\[\\simplify{(({a1+a3})x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))}\\] into partial fractions.

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Input the partial fractions here: [[0]].

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Input as the sum of partial fractions.

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Split \\[\\simplify{(({b1+b3})x^2+{b1*p+b1*q+b2+2*p*b3} * x + {b1*p*q + b2*q + b3*p^2})/ ((x + {p})^2 * (x + {q}))}\\] into partial fractions.

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Input the partial fractions here: [[0]].

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Input as the sum of partial fractions.

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Split \\[\\simplify{(({c1+c3})x^2+{c1*p1+c1*q1+c2+2*p1*c3} * x + {c1*p1*q1 + c2*q1 + c3*p1^2})/ ((x + {p1})^2 * (x + {q1}))}\\] into partial fractions.

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Input the partial fractions here: [[0]].

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Input as the sum of partial fractions.

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5/08/2012:

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Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.

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12/08/2012:

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Back to one input of a fraction and trapped input in Forbidden Strings.

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Used the except feature of ranges to get non-degenerate examples.

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Checked calculation.OK.

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Improved display in content areas.

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