Let Rz] be the set of all polynomials in variable x Let R(x) be the set of all rational functions over R, that is, R(1) := {p(¹) Note that R[x] R(x) and that R(x) is a field. : p(x), q(x) = R[x], q(x) ‡ 0 (x) #0}. Let A (r) be a 3 x 3 matrix with entires from R(x) given by G -X Dotorming if 4(r) is nonsingular/invertible over R(x), and if so, then find A-¹(x). A(x) = X 2+x -1+x² -3+3x+x² x² -1-x-x³ -2x - x² - 7³

Let f (x) = x + ax + b. How many reducible polynomials of f(x) over Z, are there?

Given a function f: ℝ → ℝ^2, for each x ∈ ℝ write f(x) = [f1(x), f2(x)] ∈ ℝ^2.Where that f1 and f2 are real valued functions of ℝ. Prove f is differentiable at a ∈ ℝ, if and only if, f1 and f2 are differentiable at a, and in that case     f'(a)  = [f1'(a)  f2'(a)] (right part of equation should be in column form)     For the following exercises, use the following definition:A function f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n if there exists a linear transformationg: ℝ^n → ℝ^m such thatlim [f(a+h)-f(a)-g(h)]/h =0h→0

Show that the function f : R – {3} → R – {5} defined by f(x) = *, is bijective and determine f-(x) for x eR – {5}.

2. (a) Show that the polynomials p, (x)=1+x, p,(x)=1+2.x – x², p;(x)=x+x² are linearly independent. (b) Decide if polynomial q(x)=-1+x is in the span{p, (x), P2 (x), P;(x)} .

Let f(x)=x6+x3+1

Find the splitting field for f (x) = (x2 + x + 2)(x2 + 2x + 2) over Z3[x]. Write f (x) as a product of linear factors.

16. Consider the function f:(-2, ∞) → [0, ∞0) defined by f (x) = vx + 2. Prove that f is injective.

Let φ1 = 1, φ2 = x, φ3 = x2 on the interval 0 ≤x ≤1. Find the following inner products:(a) < φ1,φ2 >(b) < φ2,φ3 >(c) Distance between φ1 and φ2