Question

Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?

Options

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

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

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

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

Answer 1

The repeated factor \[(x-\mathrm{a})^{2}\] requires terms for powers one and two. The distinct factor \[(x-\mathrm{b})\] requires its own constant-numerator term.