# How do I find the partial-fraction decomposition of #(2x^3+7x^2-2x+6)/(x^4+4)#?

First factor the denominator to find

Then solve

finding

It will have quadratic factors with Real coefficients.

Now attempt to solve:

Equating coefficients we get:

From (i) and (iv) we get:

Substitute these into (ii) to get:

So

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Here are the roots in the complex plane:

graph{((x-1)^2+(y-1)^2 - 0.01)((x+1)^2+(y-1)^2 - 0.01)((x-1)^2+(y+1)^2 - 0.01)((x+1)^2+(y+1)^2-0.01) = 0 [-5, 5, -2.5, 2.5]}

Now pick pairs of factors to multiply to get Real coefficients.

To do this, pick the ones which are complex conjugates:

and

For the rest of the partial fraction decomposition, see the other answers.

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To find the partial fraction decomposition of the given rational function (\frac{2x^3 + 7x^2 - 2x + 6}{x^4 + 4}), you first need to factor the denominator completely, which is (x^4 + 4). However, (x^4 + 4) does not factor further into real or rational factors. So, it remains as (x^4 + 4).

Next, you perform long division or polynomial division to rewrite the original function as (2x^3 + 7x^2 - 2x + 6) divided by (x^4 + 4). The quotient will be (0), and the remainder will be (2x^3 + 7x^2 - 2x + 6).

Since the degree of the numerator is less than the degree of the denominator, the fraction is already in its simplest form. Therefore, the partial fraction decomposition of (\frac{2x^3 + 7x^2 - 2x + 6}{x^4 + 4}) is (\frac{2x^3 + 7x^2 - 2x + 6}{x^4 + 4}).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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