<Dl> <Dd> q (x) = (x − a 1) j 1 ⋯ (x − a m) j m (x 2 + b 1 x + c 1) k 1 ⋯ (x 2 + b n x + c n) k n (\ displaystyle q (x) = (x-a_ (1)) ^ (j_ (1)) \ cdots (x-a_ (m)) ^ (j_ (m)) (x ^ (2) + b_ (1) x + c_ (1)) ^ (k_ (1)) \ cdots (x ^ (2) + b_ (n) x + c_ (n)) ^ (k_ (n))) </Dd> </Dl> <Dd> q (x) = (x − a 1) j 1 ⋯ (x − a m) j m (x 2 + b 1 x + c 1) k 1 ⋯ (x 2 + b n x + c n) k n (\ displaystyle q (x) = (x-a_ (1)) ^ (j_ (1)) \ cdots (x-a_ (m)) ^ (j_ (m)) (x ^ (2) + b_ (1) x + c_ (1)) ^ (k_ (1)) \ cdots (x ^ (2) + b_ (n) x + c_ (n)) ^ (k_ (n))) </Dd> <P> where a,..., a, b,..., b, c,..., c are real numbers with b − 4c <0, and j,..., j, k,..., k are positive integers . The terms (x − a) are the linear factors of q (x) which correspond to real roots of q (x), and the terms (x + b x + c) are the irreducible quadratic factors of q (x) which correspond to pairs of complex conjugate roots of q (x). </P> <P> Then the partial fraction decomposition of f (x) is the following: </P>

Fractional expressions with separate fractions in the numerator denominator or both are