Question

Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. (a) x − 20 x2 + x − 20 Incorrect: Your answer is incorrect. (b) x2 x2 + x + 20 Incorrect: Your answer is incorrect.

Answer 1

`(x - 20)/( x^2 + x - 20) = (5)/(9(x-4))+(4)/((x+5))`

Explanation:

Given

`(x - 20)/( x^2 + x - 20)`

Required

Decompose into partial fraction

Start by factorizing the denominator:

`x^2 + x - 20 =x^2 + 5x -4x- 20`

`x^2 + x - 20 =x(x + 5)-4(x+5)`

`x^2 + x - 20 =(x-4)(x+5)`

So, we have:

`(x - 20)/( x^2 + x - 20) = (A)/(x-4)+(B)/(x+5)`

Take LCM

`(x - 20)/( x^2 + x - 20) = (A(x+5)+B(x-4))/((x-4)(x+5))`

Cancel out the denominator

`x - 20 = A(x+5)+B(x-4)}`

Open brackets

`x - 20 = Ax+5A+Bx-4B`

Collect Like Terms

`x - 20 = Ax+Bx+5A-4B`

`x - 20 = (A+B)x+5A-4B`

By comparison;

`(A +B)x = x` ==>
`A + B = 1`

`5A - 4B = 1`

Make A the subject in
`A + B = 1`

`A = 1 - B`

Substitute 1 - B for A in
`5A - 4B = 1`

`5(1-B) -4B = 1`

`5-5B -4B = 1`

`5-9B = 1`

Collect Like Terms

`-9B = 1 - 5`

`-9B = - 4`

`B = (4)/(9)`

`A = 1 - B`

`A = 1 - (4)/(9)`

`A = (9-4)/(9)`

`A = (5)/(9)`

So:

`(x - 20)/( x^2 + x - 20) = (A)/(x-4)+(B)/(x+5)`

`(x - 20)/( x^2 + x - 20) = (5)/(9(x-4))+(4)/((x+5))`