Prove that x^8 - 14x^4 + 25 is irreducible

Expert Solution

Colby Messinger answered 1 year ago

Show that f(x) = x^4+x^3+3x^2+7x−8 is irreducible over Z.

a) Prove by induction that if a product of n polynomials is divisible by an irreducible...

a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p(x) then at least one of them is divisible by p(x). You can assume without a proof that this fact is true for two polynomials. b) Give an example of three polynomials a(x), b(x) and c(x), such that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x) nor b(x).

Question 1: a) What are the complex roots of x^8-1 ? b) What are the irreducible...

Question 1: a) What are the complex roots of x^8-1 ? b) What are the irreducible factors of f(x)=x^8-1 in R[x]? c) What are the irreducible factors of f(x)=x^8-1 in Q[x]?

Write f(x)=x^4+2x^3+2x+1 as a product of irreducible polynomials, considered as a polynomial in Z3[x], Z5[x], and...

Write f(x)=x^4+2x^3+2x+1 as a product of irreducible polynomials, considered as a polynomial in Z3[x], Z5[x], and Z7[x], respectively. 1. 2. Let f(x) be as in the previous exercise. Choose D among the polynomial rings in that exercise, so that the factor ring D/〈f(├ x)〉┤i becomes a field. Find the inverse of x+〈f├ (x)〉┤i in this field.

1. Use the derivative function, f'(x)f′(x), to determine where the function f(x)=−2x^2+14x−8 is increasing. 2.Use the...

1. Use the derivative function, f'(x)f′(x), to determine where the function f(x)=−2x^2+14x−8 is increasing. 2.Use the derivative function f'(x)f′(x) to determine where the function f(x)=2x^3−27x^2+108x+13 is increasing. 3.Use the derivative function f'(x)f′(x) to determine where the function f(x)=2x^3−27x^2+108x−12 is decreasing. 4.Find each value of the function f(x)=−x^3+12x+9 where the line tangent to the graph is horizontal. x=

For an irreducible Markov chain, either all states are positive recurrent or none are. Prove.

The cost, in dollars, of producing x belts is given by C(x)=822 + 14x - 0.075x2....

The cost, in dollars, of producing x belts is given by C(x)=822 + 14x - 0.075x2. Find the rate at which average cost is changing when 256 belts have been produced. When 256 belts have been produced, the average cost is changing at _____, dollars per belt or belt per dollars(choose one please) for each additonal belt.

Maximize p = 14x + 10y + 12z subject to x + y − z ≤...

Maximize p = 14x + 10y + 12z subject to x + y − z ≤ 3 x + 2y + z ≤ 8 x + y ≤ 5 x ≥ 0, y ≥ 0, z ≥ 0 P= (x,y,z)=

For the matrix X answer the questions below. X = ( 4 8 8 ; 3...

For the matrix X answer the questions below. X = ( 4 8 8 ; 3 6 − 9 ), this is a 2x3 matrix (a) Calculate XX’ and compute the corresponding Eigen values and Eigen vectors. (b) Obtain the singular-value decomposition of the matrix X. Note that: Eigen values for X’X are: λ1 =150, λ2 = 120, and λ3 = 0. The corresponding Eigen vectors are: eigenvector1 = 1/√30 (1 2 5), eigenvector2 = 1/√6 (1 2 -1), and...

how many irreducible polynomials of degree 2 in Z3 [x]

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