作业帮因式分解x^8+x^6+x^4+x^2+1
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/28 13:43:46
因式分解x^8+x^6+x^4+x^2+1
因式分解x^8+x^6+x^4+x^2+1
因式分解x^8+x^6+x^4+x^2+1
有两个公式 a5+b5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)
a5-b5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)
原式x^8+x^6+x^4+x^2+1=(x^2-1)(x^8+x^6+x^4+x^2+1)/(x^2-1)
=(x^10-1)/(x^2-1) =(x^5+1)(x^5-1)/(x^2-1)
=(x+1)(x^4-x^3+x^2-x+1)(x-1)(x^4+x^3+x^2+x+1)/(x^2-1)
=(x^4-x^3+x^2-x+1))(x^4+x^3+x^2+x+1)
x^8+x^6+x^4+x^2+1
=(x^2-1)(x^8+x^6+x^4+x^2+1)/(x^2-1)
=(x^10-1)/(x^2-1)
=(x^5+1)(x^5-1)/(x^2-1)
=(x+1)(x^4+x^3-x+1)(x-1)(x^4+x^3+x^2+x+1)/(x^2-1)
=(x^4+x^3-x+1)(x^4+x^3+x^2+x+1)
原式 = x^8 + x^6 + x^4 + x^2 + 1
=x^8 + 2x^6 + 3x⁴ + 2x² + 1 - x²(x⁴ + 2x² + 1)
=(x⁴+ x² + 1)² - [x(x² + 1)]²
= (x⁴- x³ + x² - x + 1)*(x⁴+ x³ + x² + x + 1)
原式x^8+x^6+x^4+x^2+1=(x^2-1)(x^8+x^6+x^4+x^2+1)/(x^2-1)
=(x^10-1)/(x^2-1) =(x^5+1)(x^5-1)/(x^2-1)
=(x+1)(x^4-x^3+x^2-x+1)(x-1)(x^4+x^3+x^2+x+1)/(x^2-1)
=(x^4-x^3+x^2-x+1))(x^4+x^3+x^2+x+1)