在实数范围内分解因式x的4次方+y的4次方

Question

在实数范围内分解因式【1】x的4次方+y的4次方 【2】x的4次方－y的4次方
化简：根号下【4－根号15】

Answer 1

x^4+y^4=x^4+2x^2y^2+y^4-2x^2y^2=(x^2+y^2)^2-2x^2y^2=(x^2+根号2xy+y^2)(x^2-根号2xy+y^2)

x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)

设 4-根号15 = (根号a-根号b)^2
则a+b=4, ab=15/4
由韦达定理构建一元二次方程 x^2-4x+15/4=0
x1=3/2, x2=5/2
所以 根号(4-根号15)=根号(5/2)-根号(3/2)=(1/2)根号(10)-(1/2)根号(6)

Answer 2

1.
x⁴+y⁴
=x⁴+2x²y²+x⁴-2x²y²
=(x²+y²)-2x²y²
=(x²+√2xy+y²)(x²-√2xy+y²)
2.
x⁴-y⁴
=(x²+y²)(x²-y²)
=(x²+y²)(x+y)(x-y)
3.
√(4-√15)
=√[(8-2√15)/2]
=√[(√5)²+(√3)²-2√3√5)/2]
=√[(√5-√3)²/2]
=(√5-√3)/√2
=(√10-√6)/2本回答被提问者采纳