How many irreducible polynomials of the form x3 + ax2 + bx + c are in...

How many irreducible polynomials of the form x³ + ax² + bx + c are in Z_p[x], such that p is a prime?

Expert Solution

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Colby Messinger answered 4 months ago

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

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).

Which of the following sets of polynomials form a basis for P2 (the space of polynomials...

Which of the following sets of polynomials form a basis for P2 (the space of polynomials of degree at most 2)? Explain. (a) {2 + 2x − x 2 , 1 + x, 3x} (b) {1 − x + 2x 2 , 2 + 4x} 1 (c) {3 + x + x 2 , −1 + x, 5 − x + x 2}

Task : (Please Answer in C#) Class Polynomials The class Polynomials is a collection of polynomials...

Task : (Please Answer in C#) Class Polynomials The class Polynomials is a collection of polynomials of either implementation stored in an instance of the generic library class List. class Polynomials { private List L; // Creates an empty list L of polynomials public Polynomials ( ) { … } // Retrieves the polynomial stored at position i in L public Polynomial Retrieve (int i) { … } // Inserts polynomial p into L public void Insert (Polynomial p) {...

Use a system of equations to find the parabola of the form y=ax^2+bx+c that goes through...

Use a system of equations to find the parabola of the form y=ax^2+bx+c that goes through the three given points. (2,-12)(-4,-60)(-3,-37)

Factor the following polynomials as a product of irreducible in the given polynomial ring. A. p(x)=2x^2+3x-2...

Factor the following polynomials as a product of irreducible in the given polynomial ring. A. p(x)=2x^2+3x-2 in Q[x] B. p(x)=x^4-9 in Q[x] C. p(x)=x^4-9in R[x] D. p(x)=x^4-9in C[x] E. f(x)=x^2+x+1 in R[x] F. f(x)=x^2+x+1in C[x]

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write...

6. Find all monic irreducible polynomials of degree ≤ 3 over Z3. Using your list, write each of the following polynomials as a product of irreducible polynomials over Z3: (a) x^4 + 2x^2 + 2x + 2. (b) 2x^3 − 2x + 1. (c) x^4 + 1.

Find the lagrange polynomials that approximate f(x) = x3 a ) Find the linear interpolation polynomial...

Find the lagrange polynomials that approximate f(x) = x3 a ) Find the linear interpolation polynomial P1(x) using the nodes x0= -1 and x1 = 0 b) Find the quadratic interpolation polynomial P2(x) using the nodes x0= -1 and x1 = 0 and x2 = 1 c) Find the cubic interpolation polynomial P3(x) using the nodes x0= -1 and x1 = 0 and x2 = 1 and x3=2 d) Find the linear interpolation polynomial P1(x) using the nodes x0= 1...

We use the form ŷ = a + bx for the least-squares line. In some computer...

We use the form ŷ = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year...

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.

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