Case III: Non-Repeated Irreducible Quadratic Factors

Case III: Non-Repeated Irreducible Quadratic Factors

A quadratic factor is irreducible if it cannot be written as a product of linear factors with real coefficients (e.g. $x^2 + x + 1$, $x^2 + 3$).

For each non-repeated irreducible factor $ax^2 + bx + c$, include a term

Example 6

Resolve $\dfrac{3x-11}{(x^2+1)(x+3)}$ into partial fractions.

Solution

$x = -3$ $\Rightarrow$ $C = -2$. Equating coefficients: $A = 2$, $B = -3$.

Hence $\dfrac{2x-3}{x^2+1} - \dfrac{2}{x+3}$.

Example 7

Resolve $\dfrac{4x^2+8x}{x^4+2x^2+9}$ into partial fractions.

Solution

$x^4 + 2x^2 + 9 = (x^2+2x+3)(x^2-2x+3)$, so

Equating coefficients yields $A = 1$, $B = 2$, $C = -1$, $D = -2$.

Hence