How do you write the partial fraction decomposition of the rational expression $x / (x² - x - 2)$ ?

Question

1517 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-22T17:15:24

$x/(x^2 - x - 2) = 1/(3(x + 1)) + 2/(3(x - 2))$

Factor the denominator:

$x^2 - x - 2 = (x + 1)(x - 2)$

Write each factor as a term with an unknown coefficient:

$x/(x^2 - x - 2) = A/(x + 1) + B/(x - 2)$

Multiply both sides by $(x + 1)(x - 2)$

$x = A(x -2) + B(x + 1)$

Make the term containing B become 0 by setting $x = -1$ :

$-1 = A(-1 -2) + B(-1 + 1)$

$-1 = A(-3)$

$A = 1/3$

$x = 1/3(x -2) + B(x + 1)$

Make the term containing $1/3$ become zero by setting $x = 2$

$2 = 1/3(2 -2) + B(2 + 1)$

$B = 2/3$

Check:

$(1/3)(1/(x + 1)) + 2/3(1/(x - 2)) =$

$(1/3)(1/(x + 1))((x - 2)/(x - 2)) + 2/3(1/(x - 2))((x + 1)/(x + 1)) =$

$(x - 2 + 2x + 2)/(3(x - 2)(x + 1)) =$

$(3x)/(3(x - 2)(x + 1)) =$

$x/((x - 2)(x + 1)) =$

$x/(x^2 - x -2)$ This checks.

How do you write the partial fraction decomposition of the rational expression x / (x² - x - 2) x / (x² - x - 2)?