Short answer, explain your reasoning (a) Find the ged in R[x] of x3 – 2x –...
2. Consider the polynomial p = x3 + x +4 € Z5 [2]. Let q = 3x +2 € Z5 [2]. (a) Is p reducible or irreducible? Prove your claim. (b) Are there any degree 2 polynomials in [g],? Explain. (c) List all degree 3 polynomials in [g]p. (d) (ungraded for thought) How many degree 4 polynomials are in (q),? Degree 5?
Preview Activity 14.1. In previous investigations, we defined irreducible polynomials and showed that irreducible polynomials in polynomial rings over fields play the same role as primes play in Z. In this investigation we will explore some methods to determine when a polynomial is irreducible, with a special emphasis on polynomials with coefficients in C, R, and Q. To begin, we will review the definition and a simple case. Let F be a field. (a) Give a formal definition of what...
Question 3 please + (20) 3. Indicate whether the reasoning of each of the following statements is correct or incorrect. Explain why or why not in each case. (Note: For an "if-then" statement, you do not need to verify that the hypothesis of the statement is true, nor come to any final conclusion ab f(x) is irreducible. Just indicate whether the conclusion correctly follows from the assumptions.) a) f(x) = +422 - 2x - 20 is irreducible in Qlx) by...
R(Xwhere the degree of R(x) is less than the degree of D(x) D(x) Use polynomial division to rewrite each expression in the form Q(x) X 8 (a) X 12 16x 9 (b) _ 2x 1 12x2 (c) x2 12 4x 1
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Write the polynomial f(x) as a product of irreducible polynomials in the given ring. Explain in each case how you know the factors are irreducible. 1) f(x) -x* + 2x2 +2x 2 in Z3[x]. 2) f(x)4 + 2x3 + 2x2 +x + 1 in Z3[x]. 3) f(x) 2x3-x2 + 3x + 2 in Q[x] 4) f(x) = 5x4-21x2 + 6x-12 in Q[x)
Please answer (c)! (a) Prove that p(x) = x3 + 9x2 + 18x + 13 is irreducible over Q. (b) Factorise 224 – 1 into monic irreducibles over Q, explaining carefully why each factor is irreducible. (c) Identify all possible isomorphisms between the following quotient rings: F5[x]/(22+x+1), F5[x]/(x² +2+2), F5[2]/(x2 + x + 3).
Find the quotient Q(x) and remainder R(x) when the polynomial P(x) is divided by the polynomial D(x). P(x) = 4x5 + 9x4 − 5x3 + x2 + x − 25; D(x) = x4 + x3 − 4x − 5 Q(x) = R(x) = Use the Factor Theorem to show that x − c is a factor of P(x) for the given values of c. P(x) = 2x4 − 13x3 − 3x2 + 117x − 135; c = −3, c = 3...
I need help with both questions. please solve and explain 2x+7 Suppose g(x) 3) Use this function to answer the following: 3+20r4 Which has the highest degree (power of x)? (circle one) Numerator/Denominator/Same Degree Without simplifying, Gm After simplifying, br Evaluate the following & give a one sentence explanation of your reasoning. lim f(x) lim f(x)= エース Does g(x) have a horizontal asymptote? If so, give its equation. If not, explain why. 1-7ォ 2TaraUse this function to answer the followinge:...