Question

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 it means for a polynomial f(x) € F[x] to be irreducible in F[0]. (b) Is the polynomial x2 – 2 irreducible in C[x]? Explain. (c) Is the polynomial x2 – 2 irreducible in R[x]? Explain. (d) Is the polynomial 22 – 2 irreducible in Q[x]? Explain. (e) The simplest non-constant polynomials are the linear (degree 1) polynomials. Show that any degree 1 polynomial in F[2] is irreducible. SEW

Answer 1

(9) Answer A non-constant polynomial polynomial with coefficients F: ciie f(nF(x)) is said to be irreducible in F[az it cannot be factored into two non-constant polynomial in F[n] of lower degree than fia) f(x) cannot be written as product of two non-constant polynomial in F[n] with degrees less than Fer). se²_2 is not irreducible in con] Because x 2-2 can be factored two non-contant polynomials in cin] 2-52 & RTVE.E C[a] & (x-2)(x+2) x?_ xr2+are+ (b) as - 2 2 2 2 :.x2_2=(x-2) catre) with x-v2, H + VE t ( [u] Hence it is not irreducible (0) x 22 not irreducible in R[u] because #2_2 = (-2)(x+2) with re-v2 #tuł ER [2]

Yes. x²_2 is irreducible in s [4] Because. xe²_2 cercatre) but we & 9 xor & om & 2 + √2 & gra] . It cannot be factored into non constant polynomicels in 9[2] Hence 22-2 is irreducible influe]

Now, M-a Linear-polynomial in ECO] can be given as f(x) = de-a with QEE since R-a EFC) itis. one factor of ne-a we factored into product of two polynomial 9 x) h(r) degreeg (n) = 1 =) degreehcnico dhin) constant AISO degree of hind=1 degree of gcn) = 0 girl is. constant. Hence it cannot be factored into two .non-constant polynomial in E[x]. x-a- is