Find the maximum common divisor of the folllwing polynomials: •f(x)=x^4 +5x^3 −4x^2 −2x and g(x)=−3x^4 −x^3 +4x^2 in Q[x]. •f(x)=2x^2 −2 and g(x)=x^4 −3x^3 +x^2 +3x−2 in R[x]


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Question Number 49238 by cesar.marval.larez@gmail.com last updated on 04/Dec/18

$F i n d t h e m a x i m u m c o m m o n d i v i s o r$ $o f t h e f o l l l w i n g p o l y n o m i a l s :$ $∙ f (x) = x^{4} + 5 x^{3} - 4 x^{2} - 2 x a n d$ $g (x) = - 3 x^{4} - x^{3} + 4 x^{2} i n Q [x] .$ $∙ f (x) = 2 x^{2} - 2 a n d g (x) = x^{4} - 3 x^{3} + x^{2} + 3 x - 2 i n R [x]$
Commented by maxmathsup by imad last updated on 04/Dec/18

$l e t f (x) = 2 x^{2} - 2 a n d g (x) = x^{4} - 3 x^{3} + x^{2} + 3 x - 2 \Rightarrow$ $g (1) = 1 - 3 + 1 + 3 - 2 = 0 a n d g (- 1) = 1 + 3 + 1 - 3 - 2 = 0 \Rightarrow x^{2} - 1 d i v i d e g (x) \Rightarrow$ $g (x) = λ (x^{2} - 1) Q (x) w i t h d e g Q = 2 w e s e e t h a t λ = 1$ $Δ (f, g) = Δ (2 (x^{2} - 1), (x^{2} - 1) Q (x)) = (x^{2} - 1) Δ (2, Q (x)) b u t$ $Q (x) = x^{2} + a x + b = 2 (\frac{1}{2} x^{2} + \frac{a}{2} x + \frac{b}{2}) = 2 v (x) \Rightarrow Δ (2, Q (x))$ $= Δ (2, 2 v (x)) = 2 Δ (1, v (x)) = 2 \Rightarrow Δ (f, g) = 2 (x^{2} - 1)$ $a n o t h e r w a y a l l r o o t s o f f a r e r o o t s o f g \Rightarrow f d i v i d e g \Rightarrow Δ (f, g) = f .$
Commented by cesar.marval.larez@gmail.com last updated on 05/Dec/18

$t h a n k u s i r$
Commented by maxmathsup by imad last updated on 05/Dec/18

$y o u a r e w e l c o m e s i r$
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