91Ó°ÊÓ

Find all the zeros of the indicated polynomial \(f(x)\) in the indicated field \(F\). $$ f(x)=x^{4}+x^{3}+3 x^{2}+2 x+2 \quad F=\mathbb{C} $$

Show that if \(S\) is a subring of \(R\), then \(S[x]\) is a subring of \(R[x]\).

Find the remainder on dividing the indicated \(f(x)\) by \(x-a\) for the indicated \(a\) in \(F[x]\) for the indicated \(F\). $$ f(x)=x^{5}+x^{3}+x^{2}+1 \quad a=-1 \quad F=\mathbb{Q} $$

In Exercises 19 through 22 construct a ring homomorphism \(\phi: Q[x] \rightarrow\) C having the indicated ideal \(K\) in \(\mathbb{Q}[x]\) as its kernel. \(K=\left\langle x^{2}-3\right\rangle\)

Find all the units in (a) \(\mathbb{Z}[x]\) (b) \(Q[x]\) (c) \(\mathbb{Z}_{5}[x]\) (d) \(\mathbb{Z}[x, y]\) (e) \(Q(x)\)

In Exercises 19 through 24 compute the product of the indicated polynomials in the indicated quotient rings. $$ 3 x+2 \text { and } 5 x-3 \quad \text { in } Q[x] /\langle x-2\rangle $$

Determine if possible, using any of the criteria given by theorems in this section, whether the indicated polynomial \(f(x)\) in \(\mathbb{Z}[x]\) is reducible over Q. Justify your answers. $$ f(x)=x^{4}-4 x^{2}+4 x-1 $$

Compute the product of the indicated polynomials in the indicated quotient rings. $$ 5 x+1 \text { and } 2 x+3 \quad \text { in } Q[x] /\left\langle x^{2}-2\right\rangle $$

Find the remainder on dividing the indicated \(f(x)\) by \(x-a\) for the indicated \(a\) in \(F[x]\) for the indicated \(F\). $$ f(x)=x^{3}+x^{2}-1 \quad a=2 \quad F=\mathbb{Z}_{5} $$

In Exercises 19 through 22 construct a ring homomorphism \(\phi: Q[x] \rightarrow\) C having the indicated ideal \(K\) in \(\mathbb{Q}[x]\) as its kernel. \(K=\left\langle x^{2}+1\right\rangle\)