Partial Fraction Proof
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Solution Summary
Some of the concepts of partial fraction are described through a few examples, to understand the concept properly.
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) ...
Education
- BEng, Allahabad University, India
- MSc , Pune University, India
- PhD (IP), Pune University, India
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