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

[therajinator]

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.

 

[haruspex]

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.