Recent content by auru

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.

I have fixed it. I'm still unsure how to find the 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)...

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 realize 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...

I've done it now, thanks!

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.