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Thread: Quick division questions

  1. #1
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    Quick division questions

    can someone help me with this,its very quick
    (3y^3 - 9y^2 - 3) divided by (3y^2 +1)

    and if anyone can quickly explain the remainder theorem that would be appreciated. thanks a lot!
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  2. #2
    Senior Member apcalculus's Avatar
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    $\displaystyle
    Quote Originally Posted by clips View Post
    can someone help me with this,its very quick
    (3y^3 - 9y^2 - 3)
    $
    Quote Originally Posted by clips View Post
    divided by $\displaystyle (3y^2 +1)$

    and if anyone can quickly explain the remainder theorem that would be appreciated. thanks a lot!

    $\displaystyle 3y^2$ goes into $\displaystyle 3y^3 $ $\displaystyle y$ times so you have:

    $\displaystyle y (3y^2 + 1) = 3y^3 + y$

    Subtract the result from the dividend:
    $\displaystyle 3y^3 - 9y^2 - 3 - (3y^3 + y) = -9y^2 - y - 3$

    Now repeat the process:
    $\displaystyle 3y^2$, the largest degree term of your divisor, goes into -$\displaystyle 9y^2$... -3 times so you have:

    $\displaystyle -3 (3y^2+1) = -9y^2 - 3$

    Subtract the result from $\displaystyle -9y^2 - y - 3$:

    $\displaystyle -9y^2 - y - 3 - (-9y^2 - 3) = -y$

    3y^2 is of greater degree than -y so -y is the remainder.

    The quotient is y-3 and the remainder is -y

    Check using:
    Dividend = quotient * divisor + remainder

    Remainder theorem:
    If you divide a polynomial p(x) by a factor (x-a), the remainder is p(a). In plain English: the remainder is the y-value you get if you evaluate the polynomial at x=a.

    example:
    If you divide $\displaystyle f(x) = x^200 + 3x^99 + 2x - 1 $ by $\displaystyle (x-1)$, the remainder is $\displaystyle f(1)$, which is $\displaystyle 1 + 3 + 2 - 1 = 5$. Note that we can conclude about the remainder without knowing the quotient.

    I hope this helps.

    Good luck!
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  3. #3
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    Quote Originally Posted by clips View Post
    can someone help me with this,its very quick
    (3y^3 - 9y^2 - 3) divided by (3y^2 +1)
    $\displaystyle 3y^3 -9y^2 -3 = (3y^2+1)(y-3)-y$

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