a) Ta có :
\(x^4-y^4=\left(x^2-y^2\right)\left(x^2+y^2\right)\)
\(=\left(x+y\right)\left(x-y\right)\left(x^2+y^2\right)\)
\(=\left(x+y\right)\left(x^3-x^2y+xy^2-y^3\right)\)
b) Ta có :
\(\left(a+b\right)^2=2\left(a^2+b^2\right)\)
\(\Rightarrow a^2+b^2+2ab=a^2+b^2+a^2+b^2\)
\(\Rightarrow a^2+b^2=2ab\)
\(\Rightarrow a^2+b^2-2ab=0\)
\(\Rightarrow\left(a-b\right)^2=0\)
\(\Rightarrow a-b=0\)
\(\Rightarrow a=b\)
Vậy ...
Ta có :
\(a^2=\left(x+y\right)^2=x^2+y^2+2xy=x^2+y^2+2b\)
\(\Rightarrow x^2+y^2=a^2-2b\)
\(a^4=\left(x+y\right)^4=x^4+C_4^1x^3y+C_4^2x^2y^2+C_4^3xy^3+y^4\)
\(\Rightarrow a^4=x^4+y^4+4x^3y+6x^2y^2+4xy^3\)
\(\Rightarrow a^4=x^4+y^4+2xy\left(2x^2+3xy+2y^2\right)\)
\(=x^4+y^4+2b\left[3b+2\left(x^2+y^2\right)\right]\)
\(=x^4+y^4+2b\left[3b+2\left(a^2-2b\right)\right]\)
\(=x^4+y^4+6b^2+4a^2b-8b\)
\(\Rightarrow x^4+y^4=a^4-\left(6b^2+4a^2b-8b\right)\)
\(=a^4-4a^2b-6b^2+8b\)