✔ 最佳答案
1.
f(X)除以(X^4-X^3-X^2+0X-1),得到商式=(X^2+1),餘式=(X^2-X+1)
∴f(X) = (X^4-X^3-X^2-1)*(X^2+1) + (X^2-X+1)
2.
因為a,b,c,d是X^4-X^3-X^2+0X-1=0的四個根
∴
(1)由根與係數關係可得
四根和:a+b+c+d = -(-1/1) = 1
兩兩乘積和 = +(-1/1) = - 1
(2)將根代入方程式可得
a^4-a^3-a^2-1=0,b^4-b^3-b^2-1=0,c^4-c^3-c^2-1=0,d^4-d^3-d^2-1=0
3.
f(a) = (a^4-a^3-a^2-1)*(a^2+1) + (a^2-a+1)
= ( 0 )*(a^2+1) + (a^2-a+1)
= a^2-a+1
同理,f(b)=b^2-b+1,f(c)=c^2-c+1,f(d)=d^2-d+1
∴f(a)+f(b)+f(c)+f(d)
= (a^2+b^2+c^2+d^2) - (a+b+c+d) + 4
= [(a+b+c+d)^2 - 2(a,b,c,d兩兩乘積和)] - (1) + 4
= [1^2 - 2*(-1)] + 3
= 6