# Partial Fraction Proof

See attachment.

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#### Solution Preview

To understand the concept, you go a step back to

P(x)/Q(x) = A(x) + R(x)/Q(x)

As per concept of division, R(x) is a remainder of P(x) for Q(x), and therefore should have at least one degree less than degree of Q(x).

e.g.,

P(x) = x^4 + x^3 + x^2 + 2 x + 1 (degree = 4)

Q(x) = x^2 + 1 ( degree = 2)

Here, P(x) can be written in terms of Q(x) as:

x^4 + x^3 + x^2 + 2 x + 1 = (x^4 + x^3 + x^2 + x) + (x + 1) == (x^2 + 1) * ( x^2 + x ) + (x +1)

You can observe that (x^2 + 1) can not further decompose (x+1)

Hence,

(x^4 + x^3 + x^2 + 2 x + 1)/(x^2 + 1) = (x^2 + x) + (x+1)/(x^2+1)

=> P(x) = Q(x) * A(x) + R(x)

Here,

P(x) = x^4 + x^3 + x^2 + 2 x + 1 (degree = 4)

Q(x) = x^2 + 1 (degree = 2)

A(x) = x^2 + x (degree = 4 - 2 = 2 == degree(P) ...

#### Solution Summary

Some of the concepts of partial fraction are described through a few examples, to understand the concept properly.