4.6.1 ANSWERS (My web server) TO EXERCISES 619 10. By successively
4.6.1 ANSWERS TO EXERCISES 619 10. By successively factoring reducible polynomials into polynomials of smaller de- gree, we must obtain a finite factorization of any polynomial into irreducibles. The factorization of the content is unique. To show that there is at most one factorization of the primitive part, the key result is to prove that if U(Z) is an irreducible factor of w(z)w(z), but not a unit multiple of the irreducible polynomial w(x), then U(Z) is a fac- tor of W(Z). This can be proved by observing that U(Z) is a factor of w(~)w(x)U(z) = T W(X) -w(z)u(z)V(z) by the result of exercise 9, where r is a nonzero constant. 11. The only row names needed would be AI, Ao, B4, Bs, Bz, BI, Bo, Cl, Co, Do. In general, let u~+~(z) = 0; then the rows needed for the proof are A,,-,, through Ao, B,,-,, through Ro, C,,-,, through CO, D,,-,, through Do, etc. 12. If nk = 0, the text s proof of (24) shows that the value of the determinant is *hk, and this equals *e;k-ll nlcjck e,”l-1(63-1) . If the polynomials have a factor of positive degree, we can artificially assume that the polynomial zero has degree zero and use the same formula with & = 0. Notes: The value R(u, w) of Sylvester s determinant is called the resultant of u and w, and the quantity (-l)deg( )(d g( )- )/2e(U)-1R( IL, u ) is called the discriminant of U, where U is the derivative of U. If U(Z) has the factored form a(z -al). . . (Z -cym), and if W(Z) = b(z -PI). . (Z -&), the resultant R(u,w) is anw(~l). . .w(ar,) = (-l) b u(&) . . . u&) = anbm n l