Question

Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials...

Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials in Z5[x], i.e. represent f(x) as a product of irreducible polynomials in Z5[x]. Demonstrate that the polynomials you obtained are irreducible.

I think i manged to factorise this polynomial.

I found a factor to be 1 so i divided the polynomial by (x-1) as its a linear factor.

So i get the form

(x3 − 2x2 + 2x − 1) = (x2-x+1)*(x-1) which is congruent to (x2+4x+1)*(x+4) (Mod 5)

Is this further reducible? Or have i factorised this polynomial wrong? Because i still get f(1) giving me 0 meaning its a linear factor

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Determine whether or not x^2+x+1 is irreducible or not over the following fields. If the polynomial...
Determine whether or not x^2+x+1 is irreducible or not over the following fields. If the polynomial is reducible, factor it. a. Z2 b. Z3 c. Z5 d. Z7
Determine the multiplicative inverse of x3 + x2 + 1 in GF(24), using the prime (irreducible)...
Determine the multiplicative inverse of x3 + x2 + 1 in GF(24), using the prime (irreducible) polynomial m(x) = x4 + x + 1 as the modulo polynomial. (Hint: Adapt the Extended Euclid’s GCD algorithm, Modular Arithmetic, to polynomials.)
Is f(x)=x^5+√6 x^4+2x^2-(3/2)x irreducible in R[x], why or why not? Is g(x)=x^3+x+1 irreducible in Z5[x] why...
Is f(x)=x^5+√6 x^4+2x^2-(3/2)x irreducible in R[x], why or why not? Is g(x)=x^3+x+1 irreducible in Z5[x] why or why not?
The polynomial f(x) given below has 1 as a zero. f(x)=x3−3x2+4x−2?
The polynomial f(x) given below has 1 as a zero. f(x)=x3−3x2+4x−2?
Is x³ + 2x² + 3x + 1 reducible over Q? If not, why not? What...
Is x³ + 2x² + 3x + 1 reducible over Q? If not, why not? What is an irreducible / primitive polynomial? What does irreducible mean, what is a prime element? Can you give an example for a primitive that is not irreducible or vice versa? What can you say about the ring R [x]? Which qualities of rings do you know, can they somehow be ordered (main ideal ring, ZPE, Euclidean)? What do the associated polynomial rings look like?...
Find the greatest common divisor of the given polynomials over the given field. Then write the...
Find the greatest common divisor of the given polynomials over the given field. Then write the greatest common divisor as a linear combination of the given polynomials. That is, given f(x) and g(x), find a(x) and b(x) so that d(x) = a(x)f(x) + b(x)g(x), where d(x) is the greatest common divisor of f(x) and g(x). (a) x^10 − x^7 − x^5 + x^3 + x^2 − 1 and x^8 − x^5 − x^3 + 1 over Q. (b) x^5 +...
Consider the function f(x) = x3 − 2x2 − 4x + 9 on the interval [−1,...
Consider the function f(x) = x3 − 2x2 − 4x + 9 on the interval [−1, 3]. Find f '(x). f '(x) = 3x2−4x−4 Find the critical values. x = Evaluate the function at critical values. (x, y) = (smaller x-value) (x, y) = (larger x-value) Evaluate the function at the endpoints of the given interval. (x, y) = (smaller x-value) (x, y) = (larger x-value) Find the absolute maxima and minima for f(x) on the interval [−1, 3]. absolute...
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials...
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials in each of the following rings, justifying your answers briefly: (i) Z3 [x]; (ii) Q[x] (this can be done easily using an appropriate theorem); (iii) R[x] (hints: you may find it helpful to write γ = 2^(1/4), the positive real fourth root of 2, and to consider factors of the form x^2 + a*x + 2^(1/2); (iv) C[x] (you may leave your answer in...
Find the derivative of the following functions (a) f(x) = ln(√x3 −2x) (b) g(x) =√x2 +...
Find the derivative of the following functions (a) f(x) = ln(√x3 −2x) (b) g(x) =√x2 + 3 x3 −5x + 1 .
Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with real coefficients. 1....
Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with real coefficients. 1. Deduce that either f(x) factors in R[x] as the product of three degree-one polynomials, or f(x) factors in R[x] as the product of a degree-one polynomial and an irreducible degree-two polynomial. 2.Deduce that either f(x) has three real roots (counting multiplicities) or f(x) has one real root and two non-real (complex) roots that are complex conjugates of each other.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT