(i) Find a non-zero polynomial in ℤ3[

(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
(i) Show that a non-zero polynomial in ??[?]Zp[x] has exactly
?−1p−1 associates.
(ii) Let ?R be a field, 0≠?(?),?(?)∈?[?]0≠a(x),b(x)∈R[x]. Prove
that ?(?)a(x) ?(?)b(x) are associates of each other if and only if
?(?)∣?(?)a(x)∣b(x)and ?(?)∣?(?)b(x)∣a(x).
Q5 (4 points) (i) Show that a non-zero polynomial in Zp[x] has exactly p - 1 associates. a(x), b(x) E R[x]. Prove that a(x) b(x) are associates of each other if and only if a(x) | b(x) (ii) Let R be a field, 0 and b(x)...
9. Show that for any non-zero real number a, the polynomial f(x)=" -a has no repeated roots in R. Hint: See 4.2.10 and 4.2.11 of the text.
Question 31 Given a zero of the polynomial function, find the remaining zeros. Then, write the polynomial as a product of linear factors i completely factored form). f(x) = x3 - x2 + 9x - 9; zero: 1 o f(x)=(x - 1)(x + 3i)(x - 3i) o f(x) = (x - 1)(x + 1)(x2 +9) O f(x) = (x + 1)(x + 3i)(x - 3i) o f(x) = (x - 1)(x + 3)(x - 3) O f(x) = (x +...
Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero fix)=x2-8x? + + 16x Determine the zero(s). If they exist. The zero(s) is/are D (Type integers or decimals. Use a comma to separate answers as needed) Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis...
(1 point) Find the polynomial of degree 9 (centered at zero) that best approximates f(x) = ln(° +5). Hint: First find a Taylor polynomial for g(x) = ln(x + 5), then use this to find the Taylor polynomial you want 1/2 Now use this polynomial to approximate L'iniz? +5) da. -1/2 Lis(z) dx =
(1 point) Find the polynomial of degree 9 (centered at zero) that best approximates f(x) 71 +23 Hint: First find a Taylor polynomial for g(2) vite then use this to find the Taylor polynomial you want. 1/2 Now use this polynomial to approximate 1 dx. 1+ 3 Do" s(2) de
For the following polynomial, one zero is given. Find the remaining zeros. The remaining zeros are (Simplify your answer. Use a comma to separate answers as needed) P(x)=x* +40X - 441, 7i is a zero Enter your answer in the answer box
Use synthetic division to decide whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k). f(x) = x2 - 18x +82; k=9-1 Is 9-i a zero of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The given k is not a zero of the polynomial function. (9-1)- (Type an exact answer, using radicals and i as...
Use the given complex zero to find the remaining three zeros for the polynomial. Write the three values as a comma separated f(x)=x^-x3-8x2-44-48 has 21 as one of its zeros.