(2) Consider the space P2(R) of all polynomials of degree at most 2, equipped with the standard polynomial addition and scalar multiplication. (c) Show that U₂ = span (P1, P2), where pi(t) = t and p₂(t) := t².

Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field of real numbers. Find the coordinates of the vector p(x) = x3 + x + 1 in the basis of P3 whose elements are the following: p0 = 1, p1 = x + a, p2 = (x+a)2  , p3 = (x+a)3, where a is an arbitrary real number.

Does there exist a polynomial of positive degree in ℤ6[x]Z6[x] that is a unit?

Find two distinct cubic polynomials over Z2 that determine the same function from Z2 to Z2.

Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic polynomial g(x) of least degree with real coefficients that has the given numbers as zeros.   -3i, 4

Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0 how do you find their intersection using matricies?

Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, andS= {1 + 2x, 1 + 3x + 5x^2 , 4x + 5x^2 } .Is Span(S)= P2?

State and prove the : The characteristic polynomial of any diagonalizable linear operator T splits.

This is a question from a linear algebra course: Let V = R[X]3, the polynomials of degree at most three, and B = {1, X, X2, X3}. Show what the image under fB is of:• the four basic elements: P1(X) = 1, P2(X) = X, P3(X) = X2 and P4(X) = X3• P(X) = 2 + 6X + 3X2 + 4X3

Let P 2 be the vector space of all polynomials of degree less than or equal to 2, under the usual vector addition and scalar multiplication. Let S = {f (x) ∈ P 2 | if deg(f ) = 2, then f has two complex roots}.  (a) Is "0 ∈ S. Why or why not? (b) Is S closed under addition? If S is closed under addition, prove it. If S is not closed under addition, provide a counterexample.   (c) Is S closed under scalar multiplication? If S is closed under scalar multiplication, prove it. If S is not closed under scalar multiplication, provide a counterexample.