Math- Partial Fractions

posted by .

Decompose the following into partial fractions after factoring the denominator as much as possible. Please show some work so I can understand how you did it.

1)x^2/((x-1)(x^2+5x+4))

2)(3x^3-5x^2+12x+4)/(x^4-16)

3)1/(x^2 (x+1)^2 )

4)(x+1)/((x^2+1) 〖(x-1)〗^2

  • EDIT -

    1) x^2
    ---------------
    (x-1)(x^2+5x+4)

    2) 3x^3-5x^2+12x+4
    -----------------
    x^4-16

    3) 1
    ----------
    x^2(x+1)^2

    4) x+1
    --------------
    (x^2+1)(x-1)^2

  • Math- Partial Fractions -

    I will do the first:
    let
    x^2/((x-1)(x+1)(x+5) = (A/(x-1) + B/(x+1) + C/(x+5)]/((x+1)(x-1)(x+5))

    then obviously the common denominator is (x-1)(x+1)(x+5)

    then only using the numerator:
    A(x+1)(x+5) + B(x-1)(x+5) + C(x-1)(x+1) = x^2
    let x = 1 --> 12A = 1 or A = 1/12
    let x = 1 --> -8B=1 , or B = -1/8
    let x = -5 --> 24C = 25 , or C = 24/25

    then x^2/((x-1)(x^2 + 5x + 4))
    = 1/(12(x-1)) - 1/((8(x+1)) + 25/(24(x+5))

  • Math- Partial Fractions -

    3) is perhaps more interesting to do, as it's already factored you can just go ahead and solve this in the usual way. But there exist a much faster way to compute partial fraction expansions, which generalizes the method used by Reiny where you insert for x the critical values to simplify the equations.

    When you have higher powers, you can in principle still use this method, by taking derivatives of both sides. But that's a bit cumbersome. What works even better is to expand the fraction around each critical point and keep only the singular terms. This is then the desired partial fraction expansion, because the difference between the function and the sum of all the expansions doesn't have any singularities anymore, it must therefore be a polynomial. However, it tends to zero at infinity, and must thus be equal to zero. This argument requires that you consider all the singularities jn the complex plane and that the numerator is of lower degree than the denominator.

    The series expansion of

    f(x) = 1/(x^2 (x+1)^2 )
    around x = 0 is:

    1/x^2 *(1 - 2 x + ...) =

    1/x^2 - 2/x + non-singular terms.

    The series expansion of

    1/(x^2 (x+1)^2 )
    around x = -1: Put x = -1+t:

    1/t^2 *1/(1-t)^2 = 1/t^2 +2/t + non-singular terms =

    1/(x+1)^2 + 2/(x+1) + non-singular terms.

    The partial fraction expansion is thus:

    1/x^2 - 2/x + 1/(x+1)^2 + 2/(x+1)

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calc easy

    Having trouble getting the correct solution. The integral of “x squared” in the numerator and “x squared plus x minus 6” in the denominator. S X2 / (X2 + x – 6) dx Thanks! That's a messy one. According to my table of integrals. …
  2. Partial Decompostion Fractions

    Can you please help me with the following questions please, I don't understand them. I know the general rule about them. Write the partial fraction decomposition of the rational expression. (x^2+4x-1)/(x^2+3)^2 (4x^3+4x^2)/(x^2+5)^2
  3. Algebra-Partial Fractions

    Can you please help me with the following questions please, I don't understand them. I know the general rule about them. Write the partial fraction decomposition of the rational expression. (x^2+4x-1)/(x^2+3)^2 (4x^3+4x^2)/(x^2+5)^2
  4. Calc II

    Perform long division on the integrand, write the proper fraction as a sum of partial fractions, and evaluate the integral: (integral of) 2y^4dy/y^3 - y^2 + y - 1 After long divison I get: (integral of)2ydy + 2(integral of)dy + (integral …
  5. Binomial

    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. …
  6. calculus

    Decompose 58-x/x^2-6x-16 into partial fractions.
  7. chemistry

    1. Divide the ions below into 2 groups, those that tend to form soluble compounds and those that tend to form insoluble compounds. (〖Pb〗^(+2), 〖Na〗^+, 〖Nh4〗^+, 〖Ag〗^+, 〖NO3〗^-, …
  8. Physics

    Coefficient of volume expansion of some liquids(a/˚C) Alcohol=1.1x〖10〗^(-4) Glycerin=5.0x〖10〗^(-4) Water=3.7x〖10〗^(-4) Ether=1.63x〖10〗^(-4) Mercury= 1.1x〖10〗^(-4) …
  9. Physics

    Coefficient of volume expansion of some liquids(a/˚C) Alcohol=1.1x〖10〗^(-4) Glycerin=5.0x〖10〗^(-4) Water=3.7x〖10〗^(-4) Ether=1.63x〖10〗^(-4) Mercury= 1.1x〖10〗^(-4 …
  10. Physics

    Coefficient of volume expansion of some liquids(a/˚C) Alcohol=1.1x〖10〗^(-4) Glycerin=5.0x〖10〗^(-4) Water=3.7x〖10〗^(-4) Ether=1.63x〖10〗^(-4) Mercury= 1.1x〖10〗^(-4 …

More Similar Questions