Question

How do you write the partial fraction decomposition of the rational expression `(x^3 - 5x + 2) / (x^2 - 8x + 15)`?

Answer 1

`(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + 45/2(1/(x - 3)) + 43/2(1/(x - 5))`

Explanation:

We need to do the division first. I am going to use long division, because I prefer it over synthetic:

.............................`x + 8`
............................__
`x² - 8x + 15)x^3 + 0x^2 - 5x + 3`
........................`-x^3 + 8x² -15x`
.......................................`8x²-20x + 3`
...................................`-8x² + 64x - 120`
.....................................................`44x - 117`

Check:

`(x + 8)(x² - 8x + 15) + 44x - 117 =`

`x³ - 8x² + 15x + 8x² -64x + 120 + 44x - 117 =`

`x³ - 5x + 3` This checks

`(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + (44x - 177)/(x² - 8x + 15)`

Now we do the decomposition on the remainder:

`(44x - 177)/(x² - 8x + 15) = A/(x - 3) + B/(x - 5)`

`44x - 177 = A(x - 5) + B(x - 3)`

Let x = 3:

`44(3) - 177 = A(3 - 5) + B(3 - 3)`

`-45 = -2A`

`A = 45/2`

Let x = 5:

`44(5) - 177 = A(5 - 5) + B(5 - 3)`

`43 = 2B`

`B = 43/2`

`(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + 45/2(1/(x - 3)) + 43/2(1/(x - 5))`