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.