Answer

问题及解答

判别下列多项式有无重因式.

Posted by haifeng on 2023-04-01 17:23:44 last update 2023-04-01 20:25:02 | Edit | Answers (2)

(1)   $f(x)=x^5-5x^4+7x^3-2x^2+4x-8$;

 

(2)   $f(x)=x^4+4x^2-4x-3$

1

Posted by haifeng on 2023-04-01 22:14:57

:mode polyn

使用 diff() 先求导

>> diff(x^5-5x^4+7x^3-2x^2+4x-8)
in> diff(x^5-5x^4+7x^3-2x^2+4x-8)
out> 5x^4-20x^3+21x^2-4x^1+4

------------------------

然后再计算 $f(x)$ 和 $f'(x)$ 的最大公因式,

>> Gcd(x^5-5x^4+7x^3-2x^2+4x-8,5x^4-20x^3+21x^2-4x^1+4)
in> Gcd(x^5-5x^4+7x^3-2x^2+4x-8,5x^4-20x^3+21x^2-4x^1+4)
49|4x^2-49x+49
------------------------

将其变为首一多项式,

>> (49|4x^2-49x+49)*(4|49)
in> (49|4x^2-49x+49)*(4|49)

out> x^2-4x^1+4
------------------------

或者使用函数 monic_polyn()

>> monic_polyn(49|4x^2-49x+49)
in> monic_polyn(49|4x^2-49x+49)
out> x^2-4x+4
------------------------

这已经说明 $f(x)$ 是有重因式的.

 

最后计算 $\frac{f(x)}{(f(x),f'(x))}$

>> (x^5-5x^4+7x^3-2x^2+4x-8)/(x^2-4x+4)
in> (x^5-5x^4+7x^3-2x^2+4x-8)/(x^2-4x+4)

out>
 quotient> q(x) = x^3-x^2-x-2
remainder> r(x) = 0

x^3-1x^2-1x^1-2
------------------------

2

Posted by haifeng on 2023-04-01 21:57:05

(2)

先求 $f(x)$ 的导函数 $f'(x)$.

>> diff(x^4+4x^3-4x-3)
in> diff(x^4+4x^3-4x-3)
out> 4x^3+12x^2-4

------------------------

>> Gcd(x^4+4x^3-4x-3,4x^3+12x^2-4)
in> Gcd(x^4+4x^3-4x-3,4x^3+12x^2-4)
-107|64
------------------------

因此 $f(x)$ 和 $f'(x)$ 互素.