Laura's copy of Partial Fractions: Repeated Factor

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Question 1

a)

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.

Input the partial fractions here: interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 8 }{ x + 9 } + \frac{ 1 }{ \left ( x + 9 \right )^{ 2 } } + \frac{ 2 }{ x - 1 }}$ .

Score: 0/2

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b)

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.

Input the partial fractions here: interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{-\frac{ 2 }{ x - 4 } + \frac{ 4 }{ \left ( x - 4 \right )^{ 2 } } + \frac{ 2 }{ x - 2 }}$ .

Score: 0/2

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c)

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.

Input the partial fractions here: interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 2 }{ x + 3 } + \frac{ 2 }{ \left ( x + 3 \right )^{ 2 } } - \frac{ 1 }{ x + 2 }}$ .

Score: 0/2

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Advice

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

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

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

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

Identifying coefficients:

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

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

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

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

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

Apply same method to solve b) and c)

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Laura's copy of Partial Fractions: Repeated Factor

a)

b)

c)

Advice