Question

Write the partial fraction decomposition of the given rational expression. startfraction 5 x plus 2 over x superscript 4 baseline minus 4 x squared endfraction

Answer 1

To decompose the rational expression startfraction 5x + 2 over x4 - 4x2 endfraction into partial fractions, we need to factor the denominator. The partial fraction decomposition of the given rational expression is startfraction A over x + B over x2 + C over (x - 2)2, where A, B, and C are the determined coefficients.

Step-by-step explanation:

To decompose the rational expression startfraction 5x + 2 over x4 - 4x2 endfraction into partial fractions, we need to factor the denominator. The denominator can be factored as x2(x2 - 4) which gives us two distinct linear factors x and x + 2, and a repeated linear factor x - 2. Therefore, we can express the rational expression as follows: startfraction A over x + B over x2 + C over (x - 2)2.

To find the values of A, B, and C, we can use the method of equating coefficients. By multiplying both sides of the equation by the common denominator, we can eliminate the fractions and solve for the unknowns. Finally, we substitute the values of A, B, and C back into the partial fraction decomposition expression.

Therefore, the partial fraction decomposition of the given rational expression is startfraction A over x + B over x2 + C over (x - 2)2, where A, B, and C are the determined coefficients.

Answer 2

Step-by-step explanation:

Given is a rational algebraic expression in x as

`(5x+2)/(x^4-4x^2)`

First let us factorize the denominator.

`x^2(x^2-4)\\=x^2(x+2)(x-2)`

Let the given term be

`(A)/(x) +(B)/(x^2) +(C)/(x-2) +(c)/(x+2) \\\\=(Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2))/(x^4-4x^2)`

Equate the numerator to get

`Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)`=
`5x+2`

Substitute x =-2

`4D (-4) = -8\\D=(1)/(2)`

Now substitute x =2

We get

`16C=12\\C=(3)/(4)`

Put x=0

`-4B=2\\B=(-1)/(2)`

Next equate X cube terms to 0 on right side

`A+C+D=0\\A=(-5)/(4)`

Hence partial fraction is

`(-1)/(2x^2) -(5)/(4x) +(1)/(2(x+2)) +(3)/(4(x-2))`