Maths Support: Partial fractions

Question 1

a)

Factorise $\simplify{ x^2 + {b+d}x + {b*d} }$

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b)

Express $\frac{\simplify{{a+c}*x+{a*d+c*b}}}{\simplify{ x^2 + {b+d}x + {b*d} }}$ in partial fractions.

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Question 2

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

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We use partial fractions to find $A$ and $B$ such that:
\[ \simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \;\;\;=\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}\]

Multiplying both sides of the equation by $\displaystyle \simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\;\;$ we obtain:

$\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \Rightarrow \simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$

Identifying coefficients:

Constant term: $\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$

Coefficent $x$: $ \simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$

On solving these equations we obtain $A = \var{a}$ and $B=\var{c}$

Which gives:\[ \simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\;\;= \simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}\]

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Maths Support: Partial fractions

Question 1

Question 1

a)

b)

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Question 2

Question 2

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