# FInd gcd of two polynomials using Euclidean Algorithm

Let $f(x)=2x^4 +3x^3 −19x^2 −28x+ 6$ and $g(x)=x^3 + 2x^2 -9x -18$ be polynomials in $\mathbb Q[x]$. Use the Euclidean Algorithm to determine the gcd in $\mathbb Q[x]$.

So far, I have the following:

$2x^4 + 3x^3 -19x^2 -28x +6 = 2x(x^3 +2x^2 -9x -18)+(-x^3 -x^2 +8x +6)$

$x^3 +2x^2 -9x -18 = 1(-x^3 -x^2 +8x +6)+(3x^2 -17x -24)$

$-x^3 -x^2 +8x +6 = [(-1/3)x](3x^2 -17x -24)+[(-20/3)x^2 +6]$

This is where I am stuck…

I'm not even sure if I am doing each step right. A little point in the right direction would be greatly appreciated. Thanks.

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Note: as a first step you want to find $f(x)=q(x)g(x)+r(x)$ where the degree of $r(x)$ is less than the degree of $g(x)$ - so you want to keep dividing through until you have a remainder which is quadratic (or less). Your basic organisation is OK, but you should be reducing the degree of the remainder at each stage - you aren't completing all the divisions. So $-x^3-x^2+8x+6$ in the first line is a cubic, and you can get it down to a quadratic. –  Mark Bennet Apr 8 '14 at 22:06
Use long division method!! –  L16H7 Apr 11 at 10:11