1. ## gcd and polynomials

Find the greatest common divisor
d(x) of
f(x) = x^5 2x^4 3x^3 15x^2 + 10x + 25
and
g(x) = x^4 6x^2 24^x 35

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using the polynomial long division

f(x) / g (x)

i ended up with 3x^3 + 135 x^2 -513x -815

couldn't factorise further !

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the step after that

x^4 6x^2 24^x 35 / 3x^3 + 135 x^2 -513x -815

and the answer was -13275x^2 -53578x+9150

but I don't know what should I do after that to get the gcd ,

and I'm not sure if my work is alright ..

any help ?

thnx

2. Hello, zis!

Find the greatest common divisor of: .$\displaystyle \begin{array}{ccc}f(x) \;=\; x^5 - 2x^4 - 3x^3 - 15x^2 + 10x + 25 \\ g(x) \;=\;x^4 - 6x^2 - 24x - 35 \end{array}$

I found that the GCD is: .$\displaystyle x^2 + 2x + 5$

Then I could factor them:

. . $\displaystyle f(x) \;=\;(x^2+2x+5)(x^3 - 4x^2 + 5)$

. . $\displaystyle g(x) \;=\;(x^2+2x+5)(x^2-2x - 7)$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In desperation, I tried the Euclidean Algorithm on the polynomials
After a slight modification, the answer popped out!

Can anyone verify that the algorithm does apply to polynomials?