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Math Help - Polynomial

  1. #1
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    Exclamation Polynomial

    how's this?
    thanks.

    y^3-gy+5
    y^3-3y-2

    and

    x^5+y^5
    x+y

    If you can show the solution, better.

    THANKS!
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  2. #2
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    Quote Originally Posted by ellenbaggao View Post
    x^5+y^5
    x+y

    If you can show the solution, better.

    THANKS!
    You could have simply tried long division

    But I am going to try a different approach...

    \frac{x^5 + y^5}{x+y} = y^4\frac{\left(\frac{x}{y}\right)^5 + 1}{\left(\frac{x}{y}\right)+1} = y^4\frac{-a^5 + 1}{-a+1}

    Where -a = \frac{x}{y}.

    You know the geometric progression formula, \frac{a^n - 1}{a - 1} = 1 + a + a^2 + a^3+ \cdots + a^{n-1}

    Thus, now we can substitute a = -\frac{x}{y}:

    y^4\frac{-a^5 + 1}{-a+1} = y^4(1 + a + a^2 + a^3 + a^4) = y^4(1 - \frac{x}{y} + \left(-\frac{x}{y}\right)^2 - \left(\frac{x}{y}\right)^3 + \left(-\frac{x}{y}\right)^4) = y^4 - xy^3+ x^2y^2 - x^3y + x^4

    So:

    \frac{x^5 + y^5}{x+y} = y^4 - xy^3+ x^2y^2 - x^3y + x^4
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  3. #3
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    Hello, ellenbaggao!

    \frac{y^3-{\color{red}g}y+5}{y^3-3y-2} . . . . What is that {\color{blue}g} ?

    \frac{x^5+y^5}{x+y}
    Factor: . \frac{(x+y)(x^4-x^3y + x^2y^2 - xy^3 + y^4)}{x+y}

    Reduce: . x^4 - x^3y + x^2y^2 - xy^3 + y^4

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  4. #4
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    hello.

    y^3-6y+5
    y^3-3y-2

    thanks.
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  5. #5
    Moo
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    Quote Originally Posted by ellenbaggao View Post
    y^3-6y+5
    y^3-3y-2

    thanks.
    note y^3-6y+5=y^3-3y-3y-2+7=(y^3-3y-2)-3y+7

    Hence \frac{y^3-6y+5}{y^3-3y-2}=\frac{(y^3-3y-2)-3y+7}{y^3-3y-2}=1+\frac{-3y+7}{y^3-3y-2}
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