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)...
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...
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...
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.