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**Unformatted text preview: **tors and irreducible quadratic factors. Once we have
this factorization of the denominator of a rational function, the next theorem tells us the form the
decomposition takes. The reader is encouraged to review the Factor Theorem (Theorem 3.6) and
its connection to the role of multiplicity to fully appreciate the statement of the following theorem.
3 We will justify this shortly. 524 Systems of Equations and Matrices
N (x)
is a rational function where the degree of N (x) less than
D (x)
and N (x) and D(x) have no common factors. Theorem 8.10. Suppose R(x) =
the degree of D(x) a • If c is a real zero of D of multiplicity m which corresponds to the linear factor ax + b, the
partial fraction decomposition includes
Am
A1
A2
+ ... +
+
2
ax + b (ax + b)
(ax + b)m
for real numbers A1 , A2 , . . . Am .
• If c is a non-real zero of D of multiplicity m which corresponds to the irreducible quadratic
ax2 + bx + c, the partial fraction decomposition includes
B1 x + C1
B 2 x + C2
Bm x + Cm
+
+ ... +
2 + bx + c
2 + bx + c)2
ax
(ax2 + bx + c)m
(ax
for real numbers...

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