How do I find a, b, c, d and e?

2

I've first tried to multiply both sides by (x^2-1) but then tried square rooting the LHS but they seem to all lead to me having to do stuff that are unnecessary and complicated. The main thing that's stumping me is having to divide the polynomial with an x^2.

_How do I find a, b, c, d and e_.jpg

Reply 1

davros

16

Original post

by BauksUnder

I've first tried to multiply both sides by (x^2-1) but then tried square rooting the LHS but they seem to all lead to me having to do stuff that are unnecessary and complicated. The main thing that's stumping me is having to divide the polynomial with an x^2.
_How do I find a, b, c, d and e_.jpg

you shouldn't need to be "square rooting" anything!

Multiply both sides by x^2 - 1 as you've suggested to clear the fractions, then you basically have (at least) 2 strategies:
compare coefficients of powers of x on both sides;
or substitute some values of x that may be helpful to evaluate the constants - e.g. x values like 0, 1 may be helpful :smile:

Reply 2

UtterlyUseless69

12

Original post

by BauksUnder

I've first tried to multiply both sides by (x^2-1) but then tried square rooting the LHS but they seem to all lead to me having to do stuff that are unnecessary and complicated. The main thing that's stumping me is having to divide the polynomial with an x^2.
_How do I find a, b, c, d and e_.jpg

I personally would either do as davros has suggested, or otherwise I might try manipulating the numerator.

If you add (x^2 - x^2) to the numerator (i.e adding 0 in a convenient way), you find is equal to 2x^4 - 2x^2 + x + 1 - x^2. The reason I did this is because now there are clearly multiples of the denominator in there that can be factored out and combined.

Factoring the 2x^4 - 2x^2 and 1 - x^2 bits, you can also say the numerator is 2x^2 (x^2 - 1) + x - (x^2 - 1).

You'll notice that two of these terms share a factor of (x^2 - 1) and so they can be combined - so how would you proceed from here?

Reply 3

Muttley79

20

Original post

by BauksUnder

I've first tried to multiply both sides by (x^2-1) but then tried square rooting the LHS but they seem to all lead to me having to do stuff that are unnecessary and complicated. The main thing that's stumping me is having to divide the polynomial with an x^2.
_How do I find a, b, c, d and e_.jpg

This is a standard question for dealing with a 'top heavy' fraction [highest power on the numerator is higher than the denominator] and splitting it into partial fractions.

Examples here:
https://www.drfrost.org/explorer.php?cuid=282791&skid=204

Reply 4

Eimmanuel

Study Forum Helper

15

Original post

by BauksUnder

I've first tried to multiply both sides by (x^2-1) but then tried square rooting the LHS but they seem to all lead to me having to do stuff that are unnecessary and complicated. The main thing that's stumping me is having to divide the polynomial with an x^2.

_How do I find a, b, c, d and e_.jpg

Hi, are you taught to do long division of polynomials?
If no, below are some resources:
1. Divisor is linear like (ax + b)
2. Divisor can be quadratic (ax² + bx + c) like your question (x² - 1)
3. Video

I would recommend that you try it to see the difference.
You can get your answer by doing two divisions.

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