How do I find the partial fraction decomposition of #(t^4+t^2+1)/((t^2+1)(t^2+4)^2)# ?

Answer 1

We can now write:

#{x^2+x+1}/{(x+1)(x+4)^2}=A/{x+1}+B/{x+4}+C/{(x+4)^2}#

By recombining the fractions,

#={A(x+4)^2+B(x+1)(x+4)+C(x+1)}/{(x+1)(x+4)^2}#

By simplifying the numertor,

#={(A+B)x^2+(8A+5B+C)x+(16A+4B+C)}/{(x+1)(x+4)#

By comparing the coefficients of the numetaors,

#A+B=1#, #8A+5B+C=1#, and #16A+4B+C=1#.
By solving the equations for #A#, #B#, and #C#,
#A=1/9#, #B=8/9#, and #C=-13/3#.
Hence, by putting #x=t^2# back in,
#{1/9}/{t^2+1}+{8/9}/{t^2+4}+{-13/3}/{(t^2+4)^2}#
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Answer 2

To find the partial fraction decomposition of the expression (t^4 + t^2 + 1) / ((t^2 + 1)(t^2 + 4)^2), follow these steps:

  1. Express the numerator as a polynomial in terms of t.

  2. Express the denominator as a product of its irreducible factors.

  3. Write the partial fraction decomposition as the sum of fractions with unknown constants over each distinct factor of the denominator.

  4. Multiply both sides of the equation by the denominator to clear the fractions.

  5. Solve for the unknown constants.

After performing these steps, you'll have the partial fraction decomposition of the given expression.

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Answer from HIX Tutor

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.

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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