Mark each of the following true or false.
__ a. The polynomial (anxn + • • • + a1x + a0) ∈ R[x] is 0 if and only if ai = 0, for i = 0, 1, • • •, n.
__ b. Tf R is a commutative ring, then R[x] is commutative.
__ c. If D is an integral domain, then D[x] is an integral domain.
__ d. If R is a ring containing divisors of 0, then R[x] has divisors of 0.
__ e. If R is a ring and f(x) and g(x) in R[x] are of degrees 3 and 4, respectively, then f(x)g(x) may be of degree 8 in R[x].
__ f. If R is any ring and f(x) and g(x) in R[x] are of degrees 3 and 4, respectively, then f(x) g(x) is always of degree 7.
__ g. If F is a subfield of E and α ∈ E is a zero of f(x) ∈ F[x], then α is a zero of h(x) = f(x) g(x) for all g(x) ∈ F[x].
__ h. If F is a field, then the units in F[x] are precisely the units in F.
__ i. If R is a ring, then X is never a divisor of 0 in R[x].
__ j. If R is a ring, then the zero divisors in R[x] are precisely the zero divisors in R.
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