求 $f(x)$ 与 $g(x)$ 的最大公因式
(1) $f(x)=x^4+x^3-3x^2-4x-1$, $g(x)=x^3+x^2-x-1$;
(2) $f(x)=x^4-4x^3+1$, $g(x)=x^3-3x^2+1$;
(3) $f(x)=x^4-10x^2+1$, $g(x)=x^4-4\sqrt{2}x^3+6x^2+4\sqrt{2}x+1$.
Answer
问题及解答
1
(1)
>> Gcd(x^4+x^3-3x^2-4x-1,x^3+x^2-x-1)
in> Gcd(x^4+x^3-3x^2-4x-1,x^3+x^2-x-1)
-3|4x-3|4
------------------------
详细步骤可使用 Gcd2()函数.
in> Gcd2(x^4+x^3-3x^2-4x-1,x^3+x^2-x-1)
out>
x^4+x^3-3x^2-4x^1-1 == (x^1)*(x^3+x^2-1x^1-1) + (-2x^2-3x^1-1)
x^3+x^2-1x^1-1 == (-1|2x^1+1|4)*(-2x^2-3x^1-1) + (-3|4x^1-3|4)
-2x^2-3x^1-1 == (8|3x^1+4|3)*(-3|4x^1-3|4)
-------------------------------------
f(x) = x^4+x^3-3x^2-4x^1-1
g(x) = x^3+x^2-1x^1-1
The remainder polynomials are:
r(1) = -2x^2-3x^1-1
r(2) = -3|4x^1-3|4
-----u(x)*f(x)+v(x)*g(x)---------
-3|4x^1-3|4 ==
(1|2x^1-1|4)*f(x) + (-1|2x^2+1|4x^1+1)*g(x)
The greatest common divisor is:
-3|4x-3|4
------------------------
2
(2)
>> Gcd(x^4-4x^3+1,x^3-3x^2+1)
in> Gcd(x^4-4x^3+1,x^3-3x^2+1)
-27|256
------------------------
>> Gcd2(x^4-4x^3+1,x^3-3x^2+1)
in> Gcd2(x^4-4x^3+1,x^3-3x^2+1)
out>
x^4-4x^3+1 == (x^1-1)*(x^3-3x^2+1) + (-3x^2-1x^1+2)
x^3-3x^2+1 == (-1|3x^1+10|9)*(-3x^2-1x^1+2) + (16|9x^1-11|9)
-3x^2-1x^1+2 == (-27|16x^1-441|256)*(16|9x^1-11|9) + (-27|256)
16|9x^1-11|9 == (-4096|243x^1+2816|243-0)*(-27|256)
-------------------------------------
f(x) = x^4-4x^3+1
g(x) = x^3-3x^2+1
The remainder polynomials are:
r(1) = -3x^2-1x^1+2
r(2) = 16|9x^1-11|9
r(3) = -27|256
-----u(x)*f(x)+v(x)*g(x)---------
-27|256 ==
(9|16x^2-333|256x^1-117|128)*f(x) + (-9|16x^3+477|256x^2+333|256x^1+207|256)*g(x)
The greatest common divisor is:
-27|256
------------------------