Partial Fractions: Numerator vs Denominator | Explained in 5:30

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In partial fraction decomposition, the degree of the numerator must be less than that of the denominator by at least one to ensure proper splitting of the expression. If the numerator's degree is equal to or greater than the denominator's, polynomial division is necessary to simplify the expression. This results in a polynomial part and a proper fraction where the numerator has a lower degree than the denominator. The numerator can have any degree up to one less than that of the denominator, and it's possible for the leading term's coefficient to be zero. Understanding this relationship is crucial for correctly applying partial fraction decomposition.
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Why in partial fractions does the power of the denominator have to be one more than that of the numerator, when splitting up the expression. Skip to 5:30. Thanks.
 
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If the numerator (N(x)) power were greater than or equal to that of the denominator (D(x)) then you could do a polynomial division to obtain N(x)/D(x) = P(x) + Q(x)/D(x), where Q has lower degree than N.
The numerator therefore has a lower degree than the denominator.
In general, it can have any degree in that range. For the purposes of calculating it, you allow it to be up to one degree less than the denominator. The coefficient of the leading term might turn out to be zero.
 

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