In the ring Q[x], consider the polynomials:
f(x) = x^5 + 2x^3 + x^2 + x + 1 and g(x) = x^4 + x^3 + x -1
a) Find the greatest common divisor d(x) of f(x) and g(x).
b) Find h(x), k(x) element in Q[x] so that h(x)f(x) + k(x)g(x) = d(x)
In the ring Q[x], consider the polynomials:
f(x) = x^5 + 2x^3 + x^2 + x + 1 and g(x) = x^4 + x^3 + x -1
a) Find the greatest common divisor d(x) of f(x) and g(x).
b) Find h(x), k(x) element in Q[x] so that h(x)f(x) + k(x)g(x) = d(x)
Let:
$\displaystyle
f(x) = x^5 + 2x^3 + x^2 + x + 1
$
$\displaystyle
g(x) = x^4 + x^3 + x -1
$
Divide f(x) by g(x) to find:
$\displaystyle
f(x) = g(x)(x-1) + 3(x^2+1)
$
Verify the remainder is zero in these divisions:
$\displaystyle
g(x) = (x^2+1)(x^2+x-1)
$
$\displaystyle
h(x) = (x^2+1)(x^3+x+1)
$
So
$\displaystyle
d(x) = (x^2+1)
$
Reinsert into the original division:
$\displaystyle
d(x) = \frac{1}{3}( f(x) - g(x)(x-1) )
$
So
$\displaystyle
h(x) = \frac{1}{3}
$
$\displaystyle
k(x) = -\frac{1}{3}(x-1)
$