Partial fractions with generating functions

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SUMMARY

The discussion focuses on expressing the rational function q(z)/p(z) where q(z) = 1 and p(z) = (1 + z)(1 + 3z) in terms of constants A and B. The user initially struggles with the conclusion that A + B = 1 and 3A + B = 0. The resolution is that two polynomials are equal if their coefficients match, leading to the equations derived from the polynomial identity. This highlights the importance of understanding polynomial coefficient comparison in partial fraction decomposition.

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  • Understanding of polynomial functions and their coefficients
  • Knowledge of partial fraction decomposition techniques
  • Familiarity with generating functions
  • Basic algebraic manipulation skills
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  • Study polynomial coefficient comparison in detail
  • Learn about advanced techniques in partial fraction decomposition
  • Explore generating functions and their applications in combinatorics
  • Review examples of rational function decomposition in calculus
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Mathematicians, students studying algebra and calculus, and anyone interested in understanding partial fractions and generating functions.

BigBoss22
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Suppose that q(z) = 1, and p(z) = (1 + z)(1 + 3z).

We wish to express q(z)/p(z) in the form

where A and B are constants. To find them, we multiply through by p(z) =
(1 + z)(1 + 3z) and obtain
1 = A(1 + 3z) + B(1 + z)
= (A + B) + (3A + B)z

Im fine up to this point, But according to my notes it is obvious that A + B = 1 and (3A + B)z = 0.

I can not see why this is the case?

Any help would be greatly appreciated.
 
Physics news on Phys.org
Never mind figured it out:

1 + 0*z = ...
 
Two polynomials are equal iff their coefficients are equal.
 

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