Question

Determine the third and fourth Taylor polynomials of
*x*^{3} + 3*x* - 1 at *x* = -1.

*p*_{3}(*x*) =

*p*_{4}(*x*) =

Answer #1

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

Consider the following.
f(x)=ln(1-x)
a) Determine the fourth Taylor polynomial of
f(x) at x = 0.
b) Use the above to estimate ln(0.6). (Give your answer correct the
four decimal places.)

Q7) Factorise the polynomial f(x) = x3 −
2x2 + 2x − 1 into irreducible polynomials in
Z5[x], i.e. represent f(x) as a product of irreducible
polynomials in Z5[x]. Demonstrate that the polynomials
you obtained are irreducible.
I think i manged to factorise this polynomial.
I found a factor to be 1 so i divided the polynomial by (x-1) as
its a linear factor.
So i get the form
(x3 − 2x2 + 2x − 1) =
(x2-x+1)*(x-1) which is...

**NUMBER THEORY**
Proof that the following polynomials do not have integer
roots.
a) x3 − x + 1
b) x3 + x2 − x + 1

Determine if the given polynomials given below span
P2
p(x) = 1 − x2 , q(x) = 1 + x, r(x) = 4x2+
3x − 1, s(x) = 3x2+ 4x + 1

1) Determine whether x3 is O(g(x)) for the following:
a. g(x) = x2 + x3
b. g(x) = x2 + x4
c. g(x) = x3 / 2 2)
Show that each of these pairs of functions are of the same
order:
a. 3x + 7, x
b. 2x2 + x - 7, x2

If V is a vector space of polynomials of degree n with real
numbers as coefficients, over R, and W is generated by
the polynomials (x 3 + 2x 2 − 2x + 1, x3 + 3x 2 − x + 4, 2x 3 +
x 2 − 7x − 7),
then is W a subspace of V , and if so, determine its basis.

1. This question is on the Taylor polynomial.
(a) Find the Taylor Polynomial p3(x) for f(x)= e^ x sin(x) about
the point a = 0.
(b) Bound the error |f(x) − p3(x)| using the Taylor Remainder
R3(x) on [−π/4, π/4].
(c) Let pn(x) be the Taylor Polynomial of degree n of f(x) =
cos(x) about a = 0. How large should n be so that |f(x) − pn(x)|
< 10^−5 for −π/4 ≤ x ≤ π/4 ?

g(x)=[x3(3x-1)^2(2x+1)]^1/2
find the derivative

Find the least upper bound and the greatest lower bound for the
two polynomials:
a) p(x) = x4 - 3x2 - 2x + 5
b) p(x) = -2x5 + 5x4 + x3 - 3x
+ 4

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