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.

Expert Solution

Colby Messinger answered 2 years ago

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]

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

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3...

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3 + 10x^2 + 16x − 11 over Q (b) f(x) = 7x^ 4 − 9x^ 2 + 6 over Q (c) f(x) = x^3 − x + 1 over Q (d) f(x) = x ^4 + 2x ^2 + 2 over Q (e) f(x) = x^ 4 + 2x^ 2 + 2 over C (f) f(x) = x ^4 + 2x^ 2 + 2...

determine if the following are homomorphisms/isomorphisms: 1. F: (Z5,+5) → (Z5,+5) where F([x]5)=[2x+1]5. 2. F :...

determine if the following are homomorphisms/isomorphisms: 1. F: (Z5,+5) → (Z5,+5) where F([x]5)=[2x+1]5. 2. F : (Z10,+10) → (Z5,+5) where F([x]10)=[2x]5. 3. F : (Z31,+31) → (Z31,+31) where F([x]31)=[7x]31.

Expand f(x) = 10 + 2x + x^4 in terms of the Legendre polynomials, showing the...

Expand f(x) = 10 + 2x + x^4 in terms of the Legendre polynomials, showing the first 5 coefficients.

What is the leading coefficient of the polynomial function f(x) = 2x+x³+4?

Find all of the real and imaginary zeros for the polynomial function. f(x)= 69 x^4-2x^3+62x^2-2x-7

f(x) = (2x − 3)(x 2 − 6) (a) Write formulas for f '(x) and f...

f(x) = (2x − 3)(x 2 − 6) (a) Write formulas for f '(x) and f ''(x). (b) Find all x-intercepts of f(x). (Exact answers, no decimals.) (c) Find all critical points of f(x). (x-values only; y-values not needed.) Classify them using the 1st or 2nd derivative test. (d) Find all inflection points of f(x). (x-values only; y-values not needed.)

1. f(x) = 1/2x-3 or y = 1/2x -3 The input is x and the output...

1. f(x) = 1/2x-3 or y = 1/2x -3 The input is x and the output is f(x) or y. What is the slope of the function? What is the y-intercept of the function? What is the x-intercept of the function? Graph the function. What is the slope of a horizontal line? Determine whether each function is increasing or decreasing? f(x) = 4x + 3 a(x) = 5 -2x k(x) = -4x +1 p(x) = 1/4x-3 q(x) = 4 4....

Find the degree 4 Taylor polynomial of f (x) = sin(2x) centered at π/6

Question