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$\frac{x^2+3}{(x+5)(x+6)}=1-\frac{11x+27}{(x+5)(x+6)}$

If you substitute $x$ with any ordinary value like $2$, you will find that value of the numerator is lower than the denominator's. So why do we still have to perform long division to get partial fractions when this is clearly not an improper fractions?

Ian Miller
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brillydev
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  • Partial fractions are important for integrating rational functions. It doesn't matter when the numerator or denominator is larger then. – Sarvesh Ravichandran Iyer Aug 21 '16 at 08:50
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    The equation is incorrect. If you substitute $x=0$ for example, the two sides aren't equal. Whether or not you get an improper fraction when substituting some value is irrelevant. – mathematician Aug 21 '16 at 09:01
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    You have misunderstood what it means for it to be an improper fraction. When your teacher talked about the numerator needing to be smaller than the denominator they meant the degree of the numerator had to be smaller than the degree of the denominator. E.g. in your example the LHS has degree of 2 for both where as on the RHS the degree of the numerator is 1 and the degree of the denominator is 2. – Ian Miller Aug 21 '16 at 09:08
  • Then what about normal fractions like 2/3 and 5/2? Aren't these proper and improper fractions? – brillydev Aug 22 '16 at 08:48

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