Question

Let f(x) g(x) and h(x) be polynomials in R[x].

Show if gcd(f(x), g(x)) = 1 and gcd(f(x), h(x)) = 1, then gcd(f(x), g(x)h(x)) = 1.

Answer #1

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

Let f(x) and g(x) be polynomials and suppose that we have f(a) =
g(a) for all real numbers a. In this case prove that f(x) and g(x)
have exactly the same coefficients. [Hint: Consider the polynomial
h(x) = f(x) − g(x). If h(x) has at least one nonzero coefficient
then the equation h(x) = 0 has finitely many solutions.]

Let
F be a field, and let f (x), g(x) ∈ F [x] be nonzero polynomials.
Then it must be the case that deg(f (x)g(x)) = deg(f (x)) +
deg(g(x)).

In the ring R[x] of polynomials with real coefficients, show
that A = {f 2 R[x] : f(0) = f(1) = 0} is an ideal.

Let R[x] be the set of all polynomials (in the variable x) with
real coefficients. Show that this is a ring under ordinary addition
and multiplication of polynomials.
What are the units of R[x] ?
I need a legible, detailed explaination

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠
0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show
that there is a real number c such that f(x) = cg(x) for all x.
(Hint: Look at f/g.)
Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be
the line tangent to the graph of g that passes through the point...

Show (-1,1)~R (where R= set of real numbers) by f(x)=
x/(1-|x|)
Use this to show g(x)=x/(d-|x|) is also a bijection (i.e. g:
(-d,d)->R)
Finally consider h(x)= x + (a+b)/2 and show it is a bijection
where h: (-d,d)->(a,b)
Conclude: R~(a,b)

Let h(x) = f (g(x) − (x^2 + 1)) . If f(0) = 3, f(2) = 5, f ' (0)
= −5, f ' (2) = 11, g(1) = 2, and g ' (1) = 4.
What is h(1) and h ' (1)?

For each of the following pairs of polynomials f(x) and g(x),
write f(x) in the form
f(x) = k(x)g(x) + r(x)
with deg(r(x)) < deg(g(x)).
a) f(x) = x^4 + x^3 + x^2 + x + 1 and g(x) = x^2 −
2x + 1.
b) f(x) = x^3 + x^2 + 1 and g(x) = x^2 − 5x + 6.
c) f(x) = x^22 − 1 and g(x) = x^5 − 1.

Show there does not exist a sequence of polynomials converging
uniformly on R to f if f(x)=cosx

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