# Abstract Algebra

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• Mar 5th 2009, 06:11 PM
Eagle3
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
• Mar 5th 2009, 07:02 PM
o_O
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: $\displaystyle \left( x + a + b\right) \Big| \left(x^3 - 3abx + a^3 + b^3\right)$
• Mar 5th 2009, 07:14 PM
Eagle3
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,
• Mar 5th 2009, 07:37 PM
o_O
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 $\displaystyle x$.
• Mar 5th 2009, 07:46 PM
Eagle3
Ok,
I think it finally clicked.

Thank you very much.