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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- Mar 5th 2009, 06:11 PMEagle3Abstract 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 PMo_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 PMEagle3
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 PMo_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 PMEagle3
Ok,

I think it finally clicked.

Thank you very much.