In April 2025
Answer to Question #301814 in Algebra for kavya
1. Find the greatest common divisor of the following polynomials over F, the field of rational numbers: (a) x 3 - 6x 2 + x + 4 and x 5 - 6x + 1. (b) x 2 + 1 and x6 + x 3 + x + 1.
(a)
"f(x)=x^3-6x^2+x+4\\\\\ng(x)=x^5-6x+1\\\\\ng:f\\\\\n\\begin{matrix}\n x^5-6x+1 &|x^3-6x^2+x+4 \\\\\n x^5-6x^4+x^3+4x^2 & x^2+6x+35\\\\\n---------\\\\\n6x^4-x^3-4x^2-6x+1\\\\\n6x^4-36x^3+6x^2+24x\\\\\n-----------\\\\\n35x^3-10x^2-30x+1\\\\\n35x^3-210x^2+35x+140\\\\\n----------\\\\\n200x^2-65x-139\n\\end{matrix}"
"r=200x^2-65x-139\\\\\nf:r"
"\\begin{matrix}\n x^3-6x^2+x+4 & |200x^2-65x-139 \\\\\n ------- & x-1135\\\\\n200x^3-1200x^2+200x+800\\\\\n200x^3-65x^2-139x\\\\\n--------\\\\\n-1135x^2+339x+800\\\\\n---------\\\\\n-22700x^2+67800x+160000\\\\\n-22700x^2+73775x+157765\\\\\n--------\\\\\n-5975x+2235\\\\\n------\\\\\n-1195x+447\n\\end{matrix}"
"r_1=-1195x+447\\\\\nr:r_1\\\\\n\\begin{matrix}\n 200x^2-65x-139 &|-1195x+447 \\\\\n -------& 200x+2345\\\\\n-23900x^2+77675x+166105\\\\\n-23900x^2+89400x\\\\\n-------\\\\\n-11725x+166105\\\\\n--------\\\\\n2345x+33221\\\\\n------\\\\\n-2802275x-39699095\\\\\n-2802275x-1048215\\\\\n-----\\\\\n-38650880\\neq0\n\\end{matrix}\\\\\ngcd(f,g)=1"
(b)
"f(x)=x^6+x^3+x+1\\\\\ng(x)=x^2+1\\\\\nf:g\\\\\n\\begin{matrix}\n x^6+x^3+x+1& |x^2+1 \\\\\n x^6+x^4 & x^4-x^2+x+1\\\\\n-----\\\\\n-x^4+x^3+x+1\\\\\n-x^4-x^2\\\\\n-----\\\\\nx^3+x^2+x+1\\\\\nx^3+x\\\\\n-----\\\\\nx^2+1\\\\\nx^2+1\\\\\n-----\\\\0\n\\end{matrix}\\\\\ngcd(f,g)=x^2+1"
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