Consider the polynomial f (x) = 125x5 − 275x4 − 2265x3 −
3213x2 − 1728x −...
Consider the polynomial f (x) = 125x5 − 275x4 − 2265x3 −
3213x2 − 1728x − 324 and c = −3/5 is a zero of multiplicity 3. find
the rest of the real zeros
Solutions
Expert Solution
Hence zeros of polynomial f(x) are:
x=-3/5 with multiplicity 3
x=6 with multiplicity 1
x=-3 with multiplicity 1
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consider f(x) = ln(x) use polynomial degree of 5!!!
a) Approximate f(0.9) and f(1.1)
b) Use Taylor remainder to find an error formula for Taylor
polynomial.
Give error bounds for each of the two approximations in (a).
Which of the two approximations in part (a) is closer to correct
value?
c) Compare an actual error in each case with error bound in part
(b).
Consider polynomial interpolation of the function
f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating
polynomial determined by m equidistant interpolation points, (2) an
interpolating polynomial determined by interpolation at the m zeros
of the Chebyshev polynomial T_m(x), and (3) by interpolating by
cubic splines instead of by a polynomial. Estimate the
approximation error by evaluation max_i |f(z_i)-p(z_i)| for many
points z_i on [-1,1]. For instance, you could use 10m points z_i.
The cubic spline interpolant can be determined in...
f(x) = x ln x
(a) Write the Taylor polynomial T3(x) for f(x) at center a =
1.
(b) Use Taylor’s inequality to give an upper bound for |R3| =
|f(x) − T3(x)| for |x − 1| ≤ 0.1. You don’t need to simplify the
number.
We’ll say a polynomial f(x) ∈ R[x] is prime if the ideal (f(x))
⊂ R[x] is prime. If F is a field with finitely many elements (e.g.,
Z/pZ), prove that f(x) ∈F [x] is prime if and only if it’s
irreducible.
Consider the polynomial f(x) = 3x 3 + 5x 2 − 58x − 40. Using
MATLAB. Find the three roots of the polynomial, i.e, x where f(x) =
0, using Newton’s method. Report the number of iterations taken by
each algorithm using a tolerance of 10−8 .
Consider the following real 3rd order polynomial
f (x)= x^3− 5.5 x^2− 5x+ 37.5
A) Use the bisection method to determine one of the roots,
employing initial guesses of xl = - 10, xu = -1, and a stopping
criterion εs=12% .
B) Use the false position method to determine a root, employing
initial guesses of xl = - 1, xu = 4, and a stopping criterion
εs=3%. Was this method the best for these initial guesses?
C) Use the...
Using MATLAB, Consider the polynomial f(x) = 3x^3 + 5x^2 − 58x −
40. Find the three roots of the polynomial, i.e, x where f(x) = 0,
using: (i) Bisection method, and (ii) Newton’s method. Report the
number of iterations taken by each algorithm using a tolerance of
10^−8 .
How does
the linear factorization of f(x), that
is,
f(x)=an
(x−c1)(x−c2)⋯(x−cn),
show that a
polynomial equation of degree n has
nroots?
Why must every
polynomial equation with real coefficients of degree 3 have at
least one real root?
If you are
given the equation of a rational function, explain how to find the
vertical asymptotes, if there is one, of the function's
graph.
If you are given the equation
of a rational function, explain how to find the horizontal
asymptote,...