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

Oct 5, 2016

(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
.....................................................$44 x - 117$

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)

$44 x - 177 = A \left(x - 5\right) + B \left(x - 3\right)$

Let x = 3:

$44 \left(3\right) - 177 = A \left(3 - 5\right) + B \left(3 - 3\right)$

$- 45 = - 2 A$

$A = \frac{45}{2}$

Let x = 5:

$44 \left(5\right) - 177 = A \left(5 - 5\right) + B \left(5 - 3\right)$

$43 = 2 B$

$B = \frac{43}{2}$

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