write the partial fraction decomposition of the following rational expression ( hint: binomial expansion with Pascal triangle can be used to expand binomials. technology may be used to solve large systems using matrices or determinants)
4
------------------------------------- x^3(x-3)(x+2)(x-1)^2(x^2+1)^2(x^2-1)^3
Look how your expression turned out.
Use brackets and such symbols as / for division to type your expression.
this better?
write the partial fraction
decomposition of the following rational expression ( hint: binomial expansion with Pascal triangle can be used to expand binomials. technology may be used to solve large systems using matrices or determinants)
4/x^3(x-3)(x+2)(x-1)^2(x^2+1)^2(x^2-1)^3
Can anyone please help me?
Are you serious?
Is this an actual question from a textbook?
Even Wolfram had a hemorrhage trying to do that one
Look at the "partial fraction expansion"
http://www.wolframalpha.com/input/?i=4%2F%28x%5E3%28x-3%29%28x%2B2%29%28x-1%29%5E2%28x%5E2%2B1%29%5E2%28x%5E2-1%29%5E3%29
Yes I am very serious. My math teacher must be playing a sick joke with us. He said it would hurt our brain. I guess he just wants us to know that there are such problems as this one that exists. thank you for your help though :)
Do you realize the number of cases you can have
e.g.
A/x + B/x^2 + C/x^3 + D/(x-3) + E/(x+2) + F(x^2+1) + G/(x^2+1)^2 + ........
and we should not forget such fractions as
?/(x(x^2+1)) or ?/(x(x-3)(x+2)(x^2-1)^3 )
can you see the absurdity of this question?
There would be 12 different factors, which would make 2^12 - 1 or 4095 subsets
I see what you're saying. Thank you very much. there is no way to do this with binomial expansion though with the pascal triangle or it's still impoosibly long?
10b-b+1
To find the partial fraction decomposition of the given rational expression, we need to decompose it into simpler fractions. The steps to do this involve factoring the denominator and determining the unknown coefficients that go with each term. Let's go through the process step by step.
Step 1: Factor the denominator
The denominator of the rational expression is:
x^3 * (x-3) * (x+2) * (x-1)^2 * (x^2+1)^2 * (x^2-1)^3
Step 2: Identify the distinct factors
From the denominator, we can identify the following distinct factors we need to consider for the partial fraction decomposition:
1. x^3
2. x-3
3. x+2
4. (x-1)^2
5. (x^2+1)^2
6. (x^2-1)^3
Step 3: Write the general form of the partial fraction decomposition
The general form of the partial fraction decomposition is:
A/x^3 + B/(x-3) + C/(x+2) + D/(x-1) + E/(x-1)^2 + (Fx+G)/(x^2+1) + (Hx+I)/(x^2+1)^2 + (Jx+K)/(x^2-1) + (Lx+M)/(x^2-1)^2 + (Nx+O)/(x^2-1)^3
Step 4: Find the unknown coefficients
To find the unknown coefficients A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, we'll need to equate the numerator of the original rational expression (in this case, 4) to the numerator of the partial fraction decomposition.
Step 5: Solve for the unknown coefficients
To solve for the unknown coefficients, we can use various techniques. For example, we can substitute specific values of x to obtain a system of linear equations or use matrix techniques to solve a large system using determinants.
Please note that calculating the specific values of the coefficients requires additional calculations and data specific to the rational expression.