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]

Expert Solution

by using definition i was solved this problem

Colby Messinger answered 1 year ago

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.

The product of (7x3 + 5x - 8)(2 - 3x) is _______ Is (x + 2) a factor of P(x) = 4x3 - 6x2 + 2x - 12? Explain.

The product of (7x3 + 5x - 8)(2 - 3x) is _______ Is (x + 2) a factor of P(x) = 4x3 - 6x2 + 2x - 12? Explain. 1.) Using synthetic substitution and the Remainder Theorem, I know that the remainder is ________ 2.) Based upon this answer in #1, and in applying the Factor Theorem, I know that (x + 2)_____ is/isn't a factor of P(x) because ______ SHOW how you can completely factor the polynomial expression x3 +...

Determine whether the polynomial is reducible or irreducible in the given polynomial ring. Justify your answers....

Determine whether the polynomial is reducible or irreducible in the given polynomial ring. Justify your answers. (c) x^4 + 1 in Z5[x] (e) 2x^3 − 5x^2 + 6x − 2 in Z[x] (f) x^4 + 4x^3 + 6x^2 + 2x + 1 in Z[x]. Hint: Substitute x − 1 for x.

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

p(x)=x^4-2x^3+x^2+2x-2 zero: 1-i use the given zero to find all the zeros of the polynomial function

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).

Factor the polynomial as a product of linear factors: x4-5x3+12x2-2x-20

Numerical Analysis: Apply Newton’s method to find the roots of polynomial P(x) = x^3 + 3x^2...

Numerical Analysis: Apply Newton’s method to find the roots of polynomial P(x) = x^3 + 3x^2 − 2x + 1. Find the convergence rate.

1)a) Factor the following polynomials completely. P(x) = x6 + 16x3 + 64, Q(x) = x4 +...

1)a) Factor the following polynomials completely. P(x) = x6 + 16x3 + 64, Q(x) = x4 + 10x2 + 25, Q(x) = x2 − 8x + 17, P(x) = x6 − 7x3 − 8, P(x) = x4 + 6x2 + 9 b)Find all its zeros. State the multiplicity of each zero. (Order your answers from smallest to largest real, followed by complex answers ordered smallest to largest real part, then smallest to largest imaginary part.

Solve for x x^4 + 2x^3 + 2x^2 + 3x + 1 = 0

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