Homework SourseCalculate by using the Euclidean Algorithm the greatest comm
Calculate by using the Euclidean Algorithm, the greatest common divisor in R[x] of x^5+5x^3+x^2+4x+1 and x^4-x^3-x-1, and write the greatest common divisor as a linear combination of the two polynomials. You may check your work with maple, but do the calculation by hand.
Solution
x^5+5x^3+x^2+4x+1 / x^4-x^3-x-1 = ( x+1 ) R ( 6x^3+2x^2+6x+2)
x^4 - x^3 -x -1 / ( 6x^3+2x^2+6x+2) = ( x/6 - 2/9 ) R ( -5x^2/9 - 5/9 )
( 6x^3+2x^2+6x+2) / ( -5x^2/9 - 5/9 ) = ( -54/5 x - 18/5 ) R 0
GCD = ( -5/9 x^2 - 18/5 )
![Calculate by using the Euclidean Algorithm, the greatest common divisor in R[x] of x^5+5x^3+x^2+4x+1 and x^4-x^3-x-1, and write the greatest common divisor as a](/WebImages/47/calculate-by-using-the-euclidean-algorithm-the-greatest-comm-1148658-1761618021-0.webp "Calculate by using the Euclidean Algorithm, the greatest common divisor in R[x] of x^5+5x^3+x^2+4x+1 and x^4-x^3-x-1, and write the greatest common divisor as a")
