1. (10 points) For each of the following statements,...

1. (10 points) For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) Let $m, d \in \mathbb{N}$ and assume that $d \mid m$. If $\alpha = a + d\mathbb{Z}$ is a square in $\mathbb{Z}/d\mathbb{Z}$, then $\tilde{\alpha} = a + m\mathbb{Z}$ is a square in $\mathbb{Z}/m\mathbb{Z}$. (b) Let $a, p, q \in \mathbb{N}$ with $p$ and $q$ odd primes. If $p \equiv q \pmod{a}$ then $\left(\frac{a}{p}\right) = \left(\frac{a}{q}\right)$.

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b) Let a,p,q e N with p and q odd primes. If p= q (mod a) then p = q (mod a^2). 2. (15 points) Let a,b,c e Z with a ≠ 0. Prove that if ax^2 + bx + c = 0 has two distinct rational roots, then a must be a perfect square. 3. (20 points) Let f(x) = x^3 - 3x^2 + 2x - 5. (a) Find all rational roots of f(x). (b) Use synthetic division to factor f(x) completely over the rational numbers. 4. (15 points) Let A = {1,2,3,4,5,6,7,8,9,10}. Define a relation R on A by aRb if and only if a + b is divisible by 3. (a) Determine whether R is reflexive, symmetric, or transitive. (b) Find the equivalence classes of R. 5. (20 points) Let f(x) = x^3 - 4x^2 + 5x - 2. (a) Use the Rational Root Theorem to find all possible rational roots of f(x). (b) Use synthetic division to find the actual rational roots of f(x). (c) Use long division to factor f(x) completely over the rational numbers. 6. (20 points) Let A = {1,2,3,4,5,6,7,8,9,10}. Define a relation R on A by aRb if and only if a^2 + b^2 is divisible by 5. (a) Determine whether R is reflexive, symmetric, or transitive. (b) Find the equivalence classes of R. 7. (20 points) Let f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1. (a) Use the Rational Root Theorem to find all possible rational roots of f(x). (b) Use synthetic division to find the actual rational roots of f(x). (c) Use long division to factor f(x) completely over the rational numbers.

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