1. Factor each of the following polynomials completely into irreducible polynomials: a) x² + 1 in Z₂[x] b) x5 + x³ + 1 in Z₂[x] c) x4 – x³ x² − x + 1 in Z3 [x] - -

Q: that 1+2i is a zero of the polynomial f with real coefficients. Which of the following statements is…

A: Given query is to find the incorrect statement.

Q: Let 0 be a root of the polynomial x³ + 3x + 7. Prove that 0 is prime in Q(0).

A: We will use the basic knowledge of polynomials and rings to answer this question properly, in…

Q: Given h(r)=−r6+12r^5−80r^4+492r^3−1611r^2+2160r−972 and 6i is a zero of h, answer the following…

A: As many parts are given , I will solve only the first 3 parts : We are given :…

Q: Derive these Legendre polynomials: P3(x) = x³ - (3/5)x P4(x) = x² - (6/7)x² + (3/35) P5(x) = x5 -…

A: Given:The Legendre polynomials:Aim:To derive the Legendre polynomial.Concept Used:Power rule of…

Q: Write the polynomial z - 2r3 -1 as product of distinct poly- nomials of degree 1 and 2.

A:

Q: Let P(x) = x6-117x3-1000 (a) Find all Zeros Of P, real and complex. to (6.) Factor R completely with…

A: Solution- Given Equation - P(x) = x6-117 x3-1000…

Q: Write g(x) = ¹ +2²³ +22² +2+2 as a product of irreducible polynomials over Z3

A:

Q: Find all the possible rational zeros of the following polynomial: f(x) = x³ – 5x² − 3x+14 O ±1, ±2,…

A:

Q: Show that each 952 MXT (a) x² + 2x³ + x + 1 polynomial is irreducible in Q[x], (b) x42x² + 8x +…

A: Please post b part complete it's not visible

Q: Determine a polynomial p(x) of degree 2 having zeros −5 and 7

A: The problem is asking us to find a polynomial of degree 2 that has zeros at -5 and 7. In other…