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Math Help - How do you do Partial Fractions with 2nd degree denomincator?

  1. #1
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    How do you do Partial Fractions with 2nd degree denomincator?

    How do you do partial fractions if the denominator has 2nd degree terms (I know the basic method)?

    My textbook says:
    Using partial fractions we get

    \frac{3s}{(s^2+4)(s^2+1)} = \frac{-s}{s^2+4} + \frac{s}{s^2+1}

    Since I have to check every single thing my textbook says (it's in my genes or something):

    \frac{3s}{(s^2+4)(s^2+1)} = \frac{A}{s^2+4} + \frac{B}{s^2+1} = \frac{(A+B)s^2+(A+4B)}{(s^2+4)(s^2+1)}

    Comparing numerator's coefficients implies that:
    A+B=0
    A+4B=0

    A=0
    B=0

    and therefore:
    \frac{3s}{(s^2+4)(s^2+1)} = \frac{0}{s^2+4} - \frac{0}{s^2+1}=0

    which is quite clearly wrong.

    What'd I do wrong? How do I get an s term in the numerator?
    Thanks.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    Quote Originally Posted by MSUMathStdnt View Post
    How do you do partial fractions if the denominator has 2nd degree terms (I know the basic method)?

    My textbook says:
    Using partial fractions we get

    \frac{3s}{(s^2+4)(s^2+1)} = \frac{-s}{s^2+4} + \frac{s}{s^2+1}

    Since I have to check every single thing my textbook says (it's in my genes or something):

    \frac{3s}{(s^2+4)(s^2+1)} = \frac{A}{s^2+4} + \frac{B}{s^2+1} = \frac{(A+B)s^2+(A+4B)}{(s^2+4)(s^2+1)}

    Comparing numerator's coefficients implies that:
    A+B=0
    A+4B=0

    A=0
    B=0

    and therefore:
    \frac{3s}{(s^2+4)(s^2+1)} = \frac{0}{s^2+4} - \frac{0}{s^2+1}=0

    which is quite clearly wrong.

    What'd I do wrong? How do I get an s term in the numerator?
    Thanks.
    Read the Procedure here:

    Partial fraction - Wikipedia, the free encyclopedia


    \frac{3s}{(s^2+4)(s^2+1)} = \frac{A \cdot s +B}{s^2+4} + \frac{C \cdot s +D}{s^2+1}

    Find A,B,C and D.
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    I'v done it like this:
    \frac{3s}{(s^2+4)\cdot(s^2+1)}=\frac{As+C}{s^2+4}+  \frac{Bs+D}{s^2+1}
    \Leftrightarrow \frac{(As+C)\cdot (s^2+1)+(Bs+D)\cdot (s^2+4)}{(s^2+4)\cdot(s^2+1)}
    \Leftrightarrow \frac{As^3+Cs^2+As+C+Bs^3+Ds^2+4Bs+4D}{(s^2+4)(s^2  +1)}
    \Leftrightarrow \frac{(A+B)s^3+(C+D)s^2+(A+4B)s+(C+D)}{(s^2+4)(s^2  +1)}

    So A+B=0 \Leftrightarrow A=-B and A+4B=3 so A=-1, B=1 and C=D=0

    So:
    \frac{3s}{(s^2+4)\cdot(s^2+1}=\frac{-s}{s^2+4}+\frac{s}{s^2+1}

    Check:
    \frac{-s}{s^2+4}+\frac{s}{s^2+1}=\frac{-s(s^2+1)+s(s^2+4)}{(s^2+4)\cdot(s^2+1)}=\frac{-s^3-s+s^3+4s}{(s^2+4)\cdot (s^2+1)}=\frac{3s}{(s^2+4)\cdot(s^2+1)}

    It seems to work.

    EDIT:
    In my opinion I don't think it's pre-algebra stuff.
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    Quote Originally Posted by Siron View Post
    I'v done it like this:
    \frac{3s}{(s^2+4)\cdot(s^2+1)}=\frac{As+C}{s^2+4}+  \frac{Bs+D}{s^2+1}
    \Leftrightarrow \frac{(As+C)\cdot (s^2+1)+(Bs+D)\cdot (s^2+4)}{(s^2+4)\cdot(s^2+1)}
    \Leftrightarrow \frac{As^3+Cs^2+As+C+Bs^3+Ds^2+4Bs+4D}{(s^2+4)(s^2  +1)}
    \Leftrightarrow \frac{(A+B)s^3+(C+D)s^2+(A+4B)s+(C+D)}{(s^2+4)(s^2  +1)}

    So A+B=0 \Leftrightarrow A=-B and A+4B=3 so A=-1, B=1 and C=D=0

    So:
    \frac{3s}{(s^2+4)\cdot(s^2+1}=\frac{-s}{s^2+4}+\frac{s}{s^2+1}

    Check:
    \frac{-s}{s^2+4}+\frac{s}{s^2+1}=\frac{-s(s^2+1)+s(s^2+4)}{(s^2+4)\cdot(s^2+1)}=\frac{-s^3-s+s^3+4s}{(s^2+4)\cdot (s^2+1)}=\frac{3s}{(s^2+4)\cdot(s^2+1)}

    It seems to work.
    Okay, but why?
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    Do you ask me why it's working? ...
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  6. #6
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    Quote Originally Posted by Siron View Post
    Do you ask me why it's working? ...
    Maybe...

    If you had to write \frac{1}{1+x^3} as sum of partial fraction, what would you do? and if it was \frac{1}{1+x^4} ?
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  7. #7
    MHF Contributor Siron's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    I would rewrite:
    \frac{1}{1+x^3}=\frac{1}{(1+x)\cdot(1-x+x^2)}=\frac{\frac{1}{3}}{1+x}+\frac{\frac{2}{3}-\frac{1}{3}x}{x^2-x+1}
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  8. #8
    MHF Contributor Siron's Avatar
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    Re: How do you do Partial Fractions with 2nd degree denomincator?

    @Also sprach zarathrusta:

    Or was the question not mentioned to me? ...
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