partial fraction $\frac{4x^2-x+6}{x^3+3x^2}$

Question

Express the following rational functions in partial fraction.

$$\frac{4x^2-x+6}{x^3+3x^2}$$

What is the form of the answer?

$\frac Ax + \frac Bx+ \frac C{x+3}$ ?

$A(x)(x+3)+B(x)(x+3)+C(x^2)$ will not get a constant term right?

Answer 1

When the factorization of the polynomial in the denominator involves a repeated term, you need to include one fraction for each power of the term up to and including the power in the factorization.

Since $x^3 + 3x^2 = x^2 (x+3)$, the form of the decomposition is $$ \frac{4x^2 - x + 6}{x^2 (x+3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+3}. $$