Question

Let
f(x) be polynomial function in field F[x].

f’(x) be the derivative of f(x).

Given the greatest common factor (f(x), f’(x))=1.

And (x-a)|f(x). Show that (x-a)^2 can not divide f(x).

Answer #1

For each polynomial f(x) ∈ Z[x], let f ' (x) denote its
derivative, which is also a polynomial in Z[x]. Let R be the
following subset of Z[x]: R = {f(x) ∈ Z[x] | f ' (0) = 0}. (a)
Prove that R is a subring of Z[x]. (b) Prove that R is not an ideal
of Z[x].

Let f be a function for which the first
derivative is f ' (x) = 2x 2 - 5 / x2 for x
> 0, f(1) = 7 and f(5) = 11. Show work for all
question.
a). Show that f satisfies the hypotheses of the Mean
Value Theorem on [1, 5]
b)Find the value(s) of c on (1, 5) that satisfyies the
conclusion of the Mean Value Theorem.

Let f(x) be a nonzero polynomial in F[x]. Show that f(x) is a
unit in F[x] if and only if f(x) is a nonzero constant polynomial,
that is, f(x) =c where 0F is not equal to c where c is a subset of
F. Hence deduce that F[x] is not a field.

Given the polynomial function f (x) = (x + 3)(x + 2)(x −1)
(a) Write all intercepts as ordered pairs
(b) Find the degree of f to determine end behavior (c) Graph the
function. Label all intercepts

Let f(x,y) be a function of x and y. The partial derivative of
f(x,y) with respect to y is equivalent to the directional
derivative of f(x,y) in the direction of the unit vector
Select one:
a. 〈0,1〉
b. 〈1,0〉
c. 〈1,1,1〉
d. 〈0,5〉

Let
F be a field and let a(x), b(x) be polynomials in F[x]. Let S be
the set of all linear combinations of a(x) and b(x). Let d(x) be
the monic polynomial of smallest degree in S. Prove that d(x)
divides a(x).

1. Let f(x)=−x^2+13x+4
a.Find the derivative f '(x)
b. Find f '(−3)
2. Let f(x)=2x^2−4x+7/5x^2+5x−9, evaluate f '(x) at x=3 rounded
to 2 decimal places.
f '(3)=
3. Let f(x)=(x^3+4x+2)(160−5x) find f ′(x).
f '(x)=
4. Find the derivative of the function f(x)=√x−5/x^4
f '(x)=
5. Find the derivative of the function f(x)=2x−5/3x−3
f '(x)=
6. Find the derivative of the function
g(x)=(x^4−5x^2+5x+4)(x^3−4x^2−1). You do not have to simplify your
answer.
g '(x)=
7. Let f(x)=(−x^2+x+3)^5
a. Find the derivative....

Determine the third Taylor polynomial of the given function at
x = 0.
f(x)=1/x+3

1 Approximation of functions by polynomials
Let the function f(x) be given by the following:
f(x) = 1/ 1 + x^2
Use polyfit to approximate f(x) by polynomials of degree k = 2,
4, and 6. Plot the approximating polynomials and f(x) on the same
plot over an appropriate domain. Also, plot the approximation error
for each case. Note that you also will need polyval to evaluate the
approximating polynomial.
Submit your code and both plots. Make sure each of...

given the polynomial function f(x)=x^2(x-3)(x+1)
find x and y intercepts of f(x) and determine whether the
graph of f crosses or touches the x-axis at each x-intercept.

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