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