In the linear-factor forms, \(a,c\ne0\) and \(ad-bc\ne0\). In the form containing \(x^2+c^2\), assume \(a\ne0\) and \(c\ne0\), so the quadratic is irreducible over the reals. For&nbsp;\(x^2+k\), assume \(k&gt;0\). All fraction identities apply where the original denominator is nonzero.
Non-repeated Linear Factors:
\(\frac{px+q}{\left( ax+b \right)\left( cx+d \right)}=\frac{A}{ax+b}+\frac{B}{cx+d}\)
Repeated Linear Factors:
\(\frac{p{{x}^{2}}+qx+r}{\left( ax+b \right){{\left( cx+d \right)}^{2}}}=\frac{A}{\left( ax+b \right)}+\frac{B}{\left( cx+d \right)}+\frac{C}{{{\left( cx+d \right)}^{2}}}\)
Non-repeated Quadratic Factor:
\(\frac{p{{x}^{2}}+qx+r}{\left( ax+b \right)\left( {{x}^{2}}+{{c}^{2}} \right)}=\frac{A}{\left( ax+b \right)}+\frac{Bx+C}{\left( {{x}^{2}}+{{c}^{2}} \right)}\)

In the linear-factor forms, \(a,c\ne0\) and \(ad-bc\ne0\). In the form containing \(x^2+c^2\), assume \(a\ne0\) and \(c\ne0\), so the quadratic is irreducible over the reals. For \(x^2+k\), assume \(k>0\). All fraction identities apply where the original denominator is nonzero.

\(\frac{px+q}{\left( ax+b \right)\left( cx+d \right)}=\frac{A}{ax+b}+\frac{B}{cx+d}\)

\(\frac{p{{x}^{2}}+qx+r}{\left( ax+b \right){{\left( cx+d \right)}^{2}}}=\frac{A}{\left( ax+b \right)}+\frac{B}{\left( cx+d \right)}+\frac{C}{{{\left( cx+d \right)}^{2}}}\)

\(\frac{p{{x}^{2}}+qx+r}{\left( ax+b \right)\left( {{x}^{2}}+{{c}^{2}} \right)}=\frac{A}{\left( ax+b \right)}+\frac{Bx+C}{\left( {{x}^{2}}+{{c}^{2}} \right)}\)

Question 114049

Question 114049

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