1. List a pair of zero divisors of (M₂(Z), +, ×):

Suppose that x = (150y +45) where x and y are integers. What are the common dividers of x and 150. What is the gcd(x,150)

17. How many times is (x – 1) a factor of the polynomial function g(x) = x* – 6x² + 8x – 3? A. None В. One С. Two D. Three E. Four

6. List all polynomials of the form x++bx+cx²+dx+e with rational coefficients that are factors of the polynomial, x¹²-1.

2. Which polynomial expresses the difference of the two polynomials below? (-8k* + 3k³ – 6) – (9k* – 11k + 2) A. –17k4 + 14k³ 3 4 - B. –17k4 + 14k³ – 8 3 O C. –17k4 – 8k³ – 8 O D. –17k4 – 8k³ – 4 -

2. Determine the greatest common divisor of a(x) = x³ – 2 and b(x) = x +1 in Q[x] and write it as a linear combination, in Q[x], of a(x) and b(x).

6. Use the Lagrange interpolating polynomial to find the polynomial passing through the points (1,2), (2,2), (4,1), (5,1).

find the Bezout coefficients and write the gcd (243, 715) a s a linear combination of 243 and 715.

The domain for variables x and y is the set {1, 2, 3}. The table below gives the values of P(x, y) for every pair of elements from the domain. For example, P(2, 3) = T because the value in row 2, column 3, is T. P 1 2 3 1 T T T Select the statement that is true. (-P(x2)^((x+y)→P(y,2))) (P(x-2)^((x+y)→→P(y,2))) (-P(2,x)^((x+y)→→P(2,y))) Oxy(P(2,x)^((x+y)→→→P(2,y))) 2 T F T 3 T IT F

4. Factor the following polynomials. a) wz? + z*wy – w²z³y b) x2 - x- 56