# [挑戰]因式分解11題(網友共解)

[挑戰]因式分解11題(網友共解)

(23)x^5 + x + 1
= x^5 + x^4 + x^3 + x^2 + x + 1 – x^4 – x^3 – x^2
= x^3(x^2 + x + 1) + (x^2 + x + 1) – x^2(x^2 + x + 1)
= (x^2 + x + 1)(x^3 + 1 – x^2)
= (x^2 + x + 1)(x^3 – x^2 + 1)

d_conan_1995解出

(24)x^7+x+1~~~此多項式在整數上不可約

[設f(x)可約成g(x)h(x)

g(x)=1最多有r的根，g(x)=-1最多也有r個根
h(x)=1最多有s的根，h(x)=-1最多也有s個根

|f(m)|=|g(m)h(m)|為質數，必有|g(m)|=1或|h(m)|=1
m有2n+1個 與上述最多2n個矛盾

f(x)=x^7+x+1

by 熊仔

http://tw.knowledge.yahoo.com/question/question?qid=1009120400825

(29)(a-b)^3+(b-c)^3+(c-a)^3= 3(a-b)(b-c)(c-a)

老王 傳音入密 說:由(28)x^3+y^3+z^3-3xya=(x+y+z)(x^2+y^2+z^2-xy-yx-zx)可解(29)

(a-b)^3+(b-c)^3+(c-a)^3-3(a-b)(b-c)(c-a)=(a-b+b-c+c-a)(———)=0

(a-b)^3+(b-c)^3+(c-a)^3= 3(a-b)(b-c)(c-a)

(25)x^8+x^7+1

=x^8+x^7+x^6-x^6+1

=x^6(x^2+x+1)-(x^6-1)

=x^6(x^2+x+1)-(x^2-1)(x^4+x^2+1)

=x^6(x^2+x+1)-(x^2-1)(x^2+x+1)(x^2-x+1)

=(x^2+x+1)[x^6-(x^2-1)(x^2-x+1)]

=(x^2+x+1)(x^6-x^4+x^3-x+1)

1. 老王

according to a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

let A=a-b,B=b-c,C=c-a

then A+B+C=0

A^3+B^3+C^3=3ABC

下一題也一樣

2. 筱紫

謝謝老王

我就沒有想到!