Why would I be embarrassed when I don't initially understand something? It may not be initially obvious to me, hence why I have asked for help.
As it turns out, in the general case I am able to divide the common divisor by a constant to attain the monic.
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...
I've just managed to do it. I expanded (x+h)^n, subtracted x^n and divided by h. Substituting 0 in for h I am left with just one term which had no h after the division, which was (n 1)x^n-1 which I now realise gives me n.x^n-1.
But I am still unsure about the others I mentioned above. I tried...
I tried the above and just came out with x^n-1. I'm not sure where to obtain the n I need.
I'm having some serious issues with [sin(x+h) - sin(x)]/h and {[1/(x+h^1/2)]-[1/(x^1/2)]}. I think the second of which can be solved with (a-b)(a+b) = a^2 - b^2 or am I completely wrong?
I don't have a very good understanding of the binomial theorem. I'm not sure what this means: "where (n|2) is the number of combinations of n items taken 2 at a time" or how it helps me.
My only experience of the binomial theorem has been making the h into a 1 thus giving me (x+h)^n =...
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.