Help me on this one :(
Express y= (7-3x-x^2)/[((1-x)^2)(2+x)] in partial fractions. Hence, prove that if x^3 and higher powers of x may be neglected, then y=(1/8)(28+30x+41x^2)
I did the first part of expressing it in partial fractions. (Since it's very difficult to type out fractions... i'll write it in terms of powers)
2(1-x)^(-1) + (1-x)^(-2) + (2+x)^(-1)
So i did a binomial on each of the three terms (fractions) above, neglecting anything after power 2. And then i added the three binomial expressions of each of the fractions, and what i got was (5 + 3x + 5.5x^2 + ...) This is different from what im supposed to prove!! Where did i go wrong?????!
I realised my mistake
I do not agree with your partial fraction sum.
To express the given expression in partial fractions, you correctly decomposed it into three fractions:
y = 2(1-x)^(-1) + (1-x)^(-2) + (2+x)^(-1)
Now, let's simplify the expression by expanding each term using the binomial theorem:
Using the binomial expansion for (1-x)^(-1), we have:
1/(1-x) = 1 + x + x^2 + ...
So, substituting this back into the expression:
2(1-x)^(-1) = 2(1 + x + x^2 + ...)
Next, using the binomial expansion for (1-x)^(-2), we have:
1/(1-x)^2 = 1 + 2x + 3x^2 + ...
So, substituting this back into the expression:
(1-x)^(-2) = 1 + 2x + 3x^2 + ...
Finally, using the binomial expansion for (2+x)^(-1), we have:
1/(2+x) = 1 - x + x^2 - x^3 + ...
So, substituting this back into the expression:
(2+x)^(-1) = 1 - x + x^2 - x^3 + ...
Now, let's combine the expansions:
2(1-x)^(-1) + (1-x)^(-2) + (2+x)^(-1)
= 2(1 + x + x^2 + ...) + (1 + 2x + 3x^2 + ...) + (1 - x + x^2 - x^3 + ...)
= (2 + 1 + 1) + (2 + 1 - 1)x + (2 + 3 + 1)x^2 + (-1 - 1 - 1)x^3 + ...
Simplifying further:
= 4 + 3x + 5x^2 - 3x^3 + ...
Now, we can see that the coefficients for x^3 and higher powers of x are not neglected. So, if we neglect x^3 and higher powers of x, the expression becomes:
y ≈ 4 + 3x + 5x^2
This is different from the expression you mentioned (1/8)(28 + 30x + 41x^2).
So, it seems like there might be an error in your calculations or simplification. It would be helpful if you could double-check your work and ensure the proper expansion and addition of terms.