1. ## 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

2. 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: $\displaystyle \left( x + a + b\right) \Big| \left(x^3 - 3abx + a^3 + b^3\right)$

3. 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,

4. 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 $\displaystyle x$.

5. Ok,
I think it finally clicked.

Thank you very much.