Exercise 2.1.2

Proof.

(a)

$$\begin{array}{llll}\hfill p& =x^{2}y+3y^2-x{y}^2+3x+x{y}^2+7x^3{y}^3& \hfill & \\ \hfill & =(7x^3)y^3+3y^2+x^{2}y+3x.& \hfill & \end{array}$$

(b)

Let $$\begin{array}{llll}\hfill q& =(y-(x_1+x_2))(y-(x_1+x_3))(y-(x_2+x_3)).& \hfill & \\ \hfill & & \hfill \end{array}$$

Consider $p=(x+x_1)(x+x_2)(x+x_3)=x^3+{\sigma }_1{x}^2+{\sigma }_{2}x+{\sigma }_3$.

Then

$$\begin{array}{llll}\hfill q& =(y-{\sigma }_1+x_3)(y-{\sigma }_1+x_2)(y-{\sigma }_1+x_1)& \hfill & \\ \hfill & =p(y-{\sigma }_1)& \hfill & \\ \hfill & ={(y-{\sigma }_1)}^3+{\sigma }_1{(y-{\sigma }_1)}^2+{\sigma }_2(y-{\sigma }_1)+{\sigma }_3& \hfill & \\ \hfill & =(y^3-3{\sigma }_1{y}^2+3{\sigma }_1^{2}y-{\sigma }_1^3)+({\sigma }_1{y}^2-2{\sigma }_1^{2}y+{\sigma }_1^3)+({\sigma }_{2}y-{\sigma }_1{\sigma }_2)+{\sigma }_3& \hfill & \\ \hfill & =y^3-2{\sigma }_1{y}^2+({\sigma }_1^2+{\sigma }_2)y+({\sigma }_3-{\sigma }_1{\sigma }_2).& \hfill & \end{array}$$