Question

Сумму x^5+y^5 выразите через сигму1 и сигму2

10 класс!​

Answer 1

$\sigma _1=x+y\; ,\; \; \sigma _2=xy\\\\S_1=x+y=\sigma _1\; ;\\\\S _2=x^2+y^2=(x+y)^2-2xy=\sigma _1^2-2\sigma _2\; ;\\\\S_3=x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)(\, (x^2+y^2)-xy)=\\\\=(x+y)(\, (x+y)^2-3xy)=\sigma _1\cdot (\sigma _1^2-3\sigma _2)=\sigma _1^3-3\sigma _1\sigma _2\; ;\\\\S_4=x^4+y^4=(x^2+y^2)^2-2x^2y^2=(\sigma _1^2-2\sigma _2)^2-2\sigma _2^2=\\\\=\sigma _1^4-4\sigma _1^2\sigma _2+2\sigma _2^2\; ;\\\\S_5=x^5+y^5\; ;\\\\\sigma _1\cdot S_4=(x+y)(x^4+y^4)=x^5+xy^4+x^4y+y^5=\\\\=(x^5+y^5)+xy(x^3+y^3)=S_5+\sigma _2\cdot S_3\; \; \Rightarrow \; \; \sigma _1\cdot S_4=S_5+\sigma _2\cdot S_3\; \; \Rightarrow$

$S_5=\sigma _1\cdot S_4-\sigma _2\cdot S_3=\sigma _1\cdot \Big ((\sigma _1^2-2\sigma _2)^2-2\sigma _2^2\Big )-\sigma _2\cdot \Big (\sigma _1\cdot (\sigma _1^2-3\sigma _2)\Big )=\\\\=\sigma _1\cdot (\sigma _1^4-4\sigma _1^2\sigma _2+2\sigma _2^2)-\sigma _2\cdot (\sigma _1^3-3\sigma _1\sigma _2)=\\\\=\sigma _1^5-5\sigma _1^3\sigma _2+5\sigma _1\sigma _2^2$