Divide x^2 + 7x + 10 by x + 2. The quotient is ax + b. What is a + b?

For f(x) = x^3 - 2x^2 + 3x - 6, what does division by x - 2 give as its remainder?

• f(2) = 0; x - 2 is a factor
• f(2) = 4; x - 2 is not a factor
• f(0) = -6; x - 2 is not a factor

Divide x^2 + 5x + 3 by x + 1. Which result is correct?

• Quotient x + 6; remainder 3
• Quotient x + 4; remainder -1
• Quotient x + 4; remainder 0

What is the remainder when x^2 - 6x + 11 is divided by x?

Divide x^3 + 2x^2 + x + 4 by x^2 + 1. What constant remainder remains?

In Q[x], the ideal generated by x^2 - 1 and x^2 + 2x + 1 has a monic generator g. What is degree(g)?

While dividing x^2 + 2x + 5 by x - 1, a student subtracts x(x - 1) and gets x + 5. What should that subtraction give?

• x + 5, because the two x terms cancel
• 3x + 5, because subtracting -x adds x
• 2x + 5, because only x^2 is subtracted
• 3x - 5, because subtraction changes the constant

Which correctly compares x^3 - 5 over Q and over R?

• It factors over Q but is irreducible over R
• It is irreducible over both coefficient fields
• It has the rational root 5 and a real linear factor
• It is irreducible over Q but has a real linear factor

In a nonzero ideal I of F[x], choose nonzero g of least degree. Why must division of any f in I by g leave remainder 0?

• A nonzero remainder would contradict minimality
• Every remainder is a nonzero constant polynomial
• The remainder lies outside I by the division theorem
• A field contains only constant polynomials

For f(x) = x^3 + 14x + 7, which prime p satisfies Eisenstein's criterion?