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Thread: Abstract Algebra

  1. #1
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    Abstract Algebra

    I really need some help getting started with this polynomial division problem. I need to find the gcd of x+a+b and x^3-3abx+a^3+b^3.

    Thanks
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  2. #2
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    Here's a link to dividing long polynomials: Polynomial Long Division

    Think of the problem as: $\displaystyle x + (a+b) \ | \ \overline{x^3 + 0x^2 - 3abx + (a^3 + b^3)}$

    You will see that you'll get a remainder of 0, indicating that: $\displaystyle \left( x + a + b\right) \Big| \left(x^3 - 3abx + a^3 + b^3\right)$
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  3. #3
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    Ok, so the first term in the quotient is x^2. I just do not know where to go with that. Do you get x^3+(a+b)x^2 and how do you subtract that from x^3+0x^2. How do you ever get the a+b without the x so you can subtract the a^3+b^3 terms? I just do not see how you get this done.

    Thanks,
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  4. #4
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    When dividing, we always match the leftmost term of each line so that when subtracting, they disappear (those in colour):

    $\displaystyle \begin{array}{cccccc}
    & & & {\color{red}x^2} & {\color{blue}-(a+b)x}& \\
    & & ---- & ---- & ---- & ---- \\
    x + (a+b) & | & {\color{red}x^3} & + 0x^2 & -3abx & + (a^3 + b^3) \\
    & & - \Big( {\color{red}x^3} & +(a+b)x^2\Big) & \\
    & & ---- & ---- & ---- \\
    & & & {\color{blue}-(a+b)x^2} & -3abx & \\
    & & & - \Big( {\color{blue}-(a+b)x^2} & - (a+b)^2x\Big)& \\
    & & & ---- & ---- & ----
    \end{array}$

    LaTex is kind of hard to work with when it comes to long division but hopefully, you can see where to go from here. Treat all the terms with $\displaystyle a$s and $\displaystyle b$s as constants in front of $\displaystyle x$.
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  5. #5
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    Ok,
    I think it finally clicked.

    Thank you very much.
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