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Partial Fractions

Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.

Irreducible Quadratics

         

If the following is an identity in \(x\): \[\frac{9x^2+9}{x^3+1}=\frac{a}{x+1}+\frac{bx+c}{x^2-x+1},\] what is the value of \(a+b+c\)?

If the following is an identity in \(x\): \[\frac{10x^2+1}{x(x^2+1)}=\frac{a}{x}+\frac{bx+c}{x^2+1},\] what is the value of \(a+b+c\)?

If the following is an identity in \(x\): \[\frac{9x-4}{x^3-2x+1}=\frac{a}{x-1}+\frac{bx+c}{x^2+x-1},\] what is the value of \(a \times b \times c\)?

If the following is an identity in \(x\): \[\frac{5x^2+Ax+B}{(7x+1)(x^2-7x+1)}=C\left(\frac{1}{7x+1}+\frac{1}{x^2-7x+1}\right),\] what is the value of \(A+B+C?\)

Which of the following is the partial fraction decomposition of

\[ \frac{ 1} { x^3 + x } ? \]

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