1 Note that the new equations are not independent, they are merely linearly independent, as they are generated by multiplication with x, the variable or the unknown.
2 The degree of a multivariate polynomial equation is given by the largest total degree of the variables, among the terms of the polynomial.
3 Note the factors at this stage carefully. They involve x, hence the resulting equation is linearly independent. Further, by virtue of their special structure, they eliminate higher powers of x.