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: <math>f(x)=\frac{x^3+16}{x^3-4x^2+8x}</math>
S'efectua la [[divisió de polinomis]] i s'obté:
After long-division, we have
: <math>f(x)=1+\frac{4x^2-8x+16}{x^3-4x^2+8x}=1+\frac{4x^2-8x+16}{x(x^2-4x+8)}</math>
SinceCom que (−4)<sup>2</sup> − 4(8) = −16 < 0, ''x''<sup>2</sup> − 4''x'' + 8 isés irreducibleirreductible, andi per sotant:
: <math>\frac{4x^2-8x+16}{x(x^2-4x+8)}=\frac{A}{x}+\frac{Bx+C}{x^2-4x+8}</math>
MultiplyingSi throughes bypassa a multiplicar ''x''<sup>3</sup> − 4''x''<sup>2</sup> + 8''x'', wea havela dreta, s'obté thela polynomialidentitat identitypolinomial:
: <math>4x^2-8x+16 = A(x^2-4x+8)+(Bx+C)x</math>
TakingPrenent ''x'' = 0, wees seepot thatveure que 16 = 8''A'', so ''A'' = 2. ComparingComparant theels coeficients de ''x''<sup>2</sup> coefficients, we sees'obté thatque 4 = ''A'' + ''B'' = 2 + ''B'', soper la qual cosa ''B'' = 2. ComparingComparant els coeficients lineals (dit d'una linearaltra coefficientsmanera, wede see''x''<sup>0</sup>) s'obté thatque −8 = −4''A'' + ''C'' = −8 + ''C'', soi per tant ''C'' = 0. Altogether,Ajuntant-ho tot:
: <math>f(x)=1+2\left(\frac{1}{x}+\frac{x}{x^2-4x+8}\right)</math>
The following example illustrates almost all the "tricks" one would need to use short of consulting a [[computer algebra system]]'''.
=== Exemple 3 ===
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