Partial Fraction: Simplifying 3/(x^3 + 3) - Urgent Help Needed

[saltrock]

Partial Fraction!Really Really Urgent

Hey I am stuck in this question.Plese help me do this

3/(x^3 +3 )

change this into partial fraction

 

[shmoe]

You first need to factor x^3+3 into irreducibles. Hint-it's the sum of 2 cubes, x and 3^(1/3).

 

[tacman]

Hi there,

I understand that you are stuck on this question and need urgent help. Don't worry, I will try my best to explain how to simplify this expression using partial fractions.

First, let's factor the denominator, x^3 + 3. We can rewrite it as (x+1)(x^2-x+1). Now, we need to find the partial fraction decomposition of 3/(x^3 + 3), which means we need to express this fraction as a sum of simpler fractions.

To do this, we need to find the constants A, B, and C such that:

3/(x^3 + 3) = A/(x+1) + Bx+C/(x^2-x+1)

To find A, we multiply both sides by (x+1) and then substitute x=-1:

3 = A(x+1)(x^2-x+1)

Substituting x=-1, we get:

3 = A(0)(3)

Therefore, A = 1.

Next, to find B and C, we need to multiply both sides by x^2-x+1 and then substitute x=0 and x=1:

3 = Bx(x^2-x+1) + C(x^3+3)

Substituting x=0, we get:

3 = C(3)

Therefore, C = 1.

Substituting x=1, we get:

3 = B(1)(1)

Therefore, B = 3.

Now that we have found the values of A, B, and C, we can rewrite the original fraction as:

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

This is the simplified form of the original expression using partial fractions. I hope this helps you understand how to solve this type of problem. If you have any further questions, please don't hesitate to ask. Good luck!