Partial Fraction decomposition (degrees) Let us say that I want to decompose the fraction 
$$\frac{x-3}{x^2 +6x+5}$$ into partial fractions. 
I know that we have to factor the denominator and write it as $$\frac{A}{x+1} + \frac{B}{x+5}.$$ Then we get
$$x - 3 = A(x+5) + B(x+1).$$
Select convenient values for $x$ and solve for $A$ and $B$. 
My question, why if the denominator is linear, the numerator has to be 1 degree less than the denominator? Not just in this specific example, but in all questions - THE top has to be one degree less than the bottom. Why? 

If $f(x)=\frac{P(x)}{Q(x)}$ is a rational function and $\deg(P(x))\ge\deg(Q(x))$ we call $f$ an Improper rational function, it is said to be "top-heavy". 
Conversely, if $\deg(P(x))\lt\deg(Q(x))$ then $f$ is a Proper rational function. 
When $\deg(P(x))\ge\deg(Q(x))$, we use polynomial long division to turn a top-heavy rational function $f(x)=\frac{P(x)}{Q(x)}$ into a proper rational function:
$$f(x)=S(x) + \frac {R(x)}{Q(x)}$$ 
where $S(x)$ and $R(x)$ are polynomials and $\deg(R(x))\lt\deg(Q(x))$.

Select convenient values for $x$ and solve for $A$ and $B$.

Finding $A$ and $B$ (and often $C$, $D$, $E$ and so on) is extremely important when you get to integral calculus if you need to evaluate integrals involving rational functions such as this one using the function from your example:
$$\int \frac{x-3}{x^2 +6x+5} \, dx$$
.
