Question

Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?

Options

• \[\frac{\mathrm{A}}{(x-\mathrm{a})^{2}}+\frac{\mathrm{B}}{x^{2}+\mathrm{b}x+\mathrm{c}}\]

• \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}}{x^{2}+\mathrm{b}x+\mathrm{c}}\]

• \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}x+\mathrm{C}}{x^{2}+\mathrm{b}x+\mathrm{c}}\]

• \[\frac{\mathrm{A}x+\mathrm{B}}{x-\mathrm{a}}+\frac{\mathrm{C}}{x^{2}+\mathrm{b}x+\mathrm{c}}\]

Answer 1

The factor \[(x-\mathrm{a})\] uses a constant numerator \[\mathrm{A}\]. The irreducible quadratic factor uses the linear numerator \[\mathrm{B}x+\mathrm{C}\].