# Search results

1. ### Monic Greatest Common Divisor

Homework Statement Find the monic greatest common divisor of two polynomials a = 6x6 + 12x5 - 6x4 -12x +12 and b = 3x4 - 3. Homework Equations The Euclidean Algorithm. The Attempt at a Solution Applying the Euclidean Algorithm, I have a = 6x6 + 12x5 - 6x4 -12x +12 = (3x4 - 3)(2x2 + 4x -2)...
2. ### Finding Units Modular Arithmetic

Homework Statement I am required to find the units of ℤ8. Homework Equations I have that ##\bar{a}## = [a]n = { a + kn, k ∈ ℤ } ##u## ∈ ℤn is a unit if ##u## divides ##\bar{1}##. The Attempt at a Solution I'm not sure how to go about this. My lecturer wrote out the multiplication table...
3. ### Differentiate from first principles

Homework Statement Differentiate from first principles with respect to x: x^n (where n belongs to the natural numbers). Homework Equations f'(x) = Lim x→0 [f(x+h) - f(x)]/h The Attempt at a Solution f'(x) = Lim x→0 [f(x+h) - f(x)]/h = Lim x→0 [(x+h)^n - x^n]/h I need some...
4. ### Differentiate from first principles

Homework Statement Differentiate (x+1)^1/2 from first principles with respect to x. Homework Equations f'(x) = Lim h→0 [f(x+h) - f(x)]/h The Attempt at a Solution f'(x) = Lim h→0 [f(x+h) - f(x)]/h f'(x) = Lim h→0 [(x+h+1)^1/2 - (x+1)^1/2]/h I'm unsure how to simplify from there.