Question

Theorem. Suppose f(x) = a_ nx^(n)+a_n−1x^(n−1)+...+a_0 is a polynomial of degree n > 0 with integer coefficients. Then f(x) is a composite number for infinitely many integers x.

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Answer #1

Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer
coefficients with an ? 0 ? a0 and there are relatively prime
integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint:
Clear denominators.]

Let f ∈ Z[x] be a nonconstant polynomial with the property that
all the roots (in comlex plane) for the equation f(x) = 0 are
distinct. Prove that there exist infinitely many positive integers
n such that f(n) is not a perfect square. Could you explain it in
number theory instead of some deep math like sigel theorem

Let f ∈ Z[x] be a nonconstant polynomial with the property that
all the roots (in comlex plane) for the equation f(x) = 0 are
distinct. Prove that there exist infinitely many positive integers
n such that f(n) is not a perfect square.

Find the Maclaurin polynomial (c = 0) of degree n = 6 for f(x) =
cos(2x). Use a calculator to compare the polynomial evaluated at
π/8 and cos(2π/8)

1. Find T5(x): Taylor polynomial of degree 5 of the function
f(x)=cos(x) at a=0.
T5(x)=
Using the Taylor Remainder Theorem, find all values of x
for which this approximation is within 0.00054 of the right answer.
Assume for simplicity that we limit ourselves to |x|≤1.
|x|≤ =
2. Use the appropriate substitutions to write down the first
four nonzero terms of the Maclaurin series for the binomial:
(1+7x)^1/4
The first nonzero term is:
The second nonzero term is:
The third...

1.)Find T5(x), the degree 5 Taylor polynomial of the function
f(x)=cos(x) at a=0.
T5(x)=
Find all values of x for which this approximation is within
0.003452 of the right answer. Assume for simplicity that we limit
ourselves to |x|≤1.
|x|≤
2.) (1 point) Use substitution to find the Taylor series of
(e^(−5x)) at the point a=0. Your answers should not include the
variable x. Finally, determine the general term an in
(e^(−5x))=∑n=0∞ (an(x^n))
e^(−5x)= + x + x^2
+ x^3 + ... = ∑∞n=0...

Suppose f(x)=x6+3x+1f(x)=x6+3x+1. In this problem, we will show
that ff has exactly one root (or zero) in the interval
[−4,−1][−4,−1].
(a) First, we show that f has a root in the interval
(−4,−1)(−4,−1). Since f is a SELECT ONE!!!! (continuous)
(differentiable) (polynomial) function on the
interval [−4,−1] and f(−4)= ____?!!!!!!!
the graph of y=f(x)y must cross the xx-axis at some point in the
interval (−4,−1) by the SELECT ONE!!!!!! (intermediate value
theorem) (mean value theorem) (squeeze theorem) (Rolle's theorem)
.Thus, ff...

1. Given an
n-element array A, Algorithm X executes an
O(n)-time computation for each even
number in A and an O(log n)-time computation for
each odd number in A.
What is the best-case running time of Algorithm X?
What is the worst-case running time of Algorithm X?
2. Given an array,
A, of n integers, give an O(n)-time algorithm that finds
the longest subarray of A such that all the numbers in that
subarray are in sorted order. Your algorithm...

1. Suppose the equation p(x,y)=-2x^2+80x-3y^2+90+100 models
profit when x represents the number of handmade chairs and y is the
number of handmade rockers produced per week.
(1) How many chairs and how many rockers will give the maximum
profit when there is not constraint?
(2) Due to an insufficient labor force they can only make a
total of 20 chairs and rockers per week (x + y = 20). So how many
chairs and how many rockers will give the...

1. Complete the PDF.
x
P(X =
x)
x · P(X =
x)
0
0.1
1
0.4
2
3
0.2
Part (a) Find the probability that X = 2.
Part (b) Find the expected value.
2. A school newspaper reporter decides to randomly survey 16
students to see if they will attend Tet (Vietnamese New Year)
festivities this year. Based on past years, she knows that 21% of
students attend Tet festivities. We are interested in the number of
students...

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