Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

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Colby Messinger answered 2 years ago

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q...

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q such that x < q < y. Strategy for solving the problem Show that there exists an n ∈ N+ such that 0 < 1/n < y - x. Letting A = {k : Z | k < ny}, where Z denotes the set of all integers, show that A is a non-empty subset of R with an upper bound in R. (Hint: Use...

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Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...

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Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

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Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q...

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