Using the division algorithm, compute the quotient and remainder of division by \(f\) by \(g\).

1. \(f=2x^4-3x^2-3,\ g=x-3\).
2. \(f=x^3+4x^2-2x+1,\ g=2x^2-2\).

Compute the following greatest common divisors.

1. \(\gcd(x^4-2x+1,x^2+2x-3)\).
2. \(\gcd(x^3-3x-2, x^3-x^2-x+1)\).
3. \(\gcd(x^6-1,x^4-1,x^3-1)\).

Reorder each of the following polynomials using the lex, grlex, and grevlex monomial orders.

1. \(x^3-4x+x^4-3x^7\).
2. \(x^2y + 4xyz^2 + 2x^2yz^2 + x^3yz + 4\).

Determine whether the polynomial \(f\) is an element of the ideal \(I\).

1. \(f = x^3-8,\ I=\langle x-2 \rangle\).
2. \(f = x^3-4x^2+x+6,\ I=\langle x^2-5x+6 \rangle\).
3. \(f = x^2,\ I=\langle x^3-x, x^2+4x \rangle\).
4. \(f = x,\ I=\langle x^3-x^2, x^4+4x^3 \rangle\).
5. \(f = x^3-7x+6,\ I=\langle x^2-5x+6, x^2-4 \rangle\).
6. \(f = x^3-3x^2-x+3,\ I=\langle x^2-5x+6, x^2-4 \rangle\).

Compute the remainder of \(f\) on division by the ordered set \(F\) using the division algorithm with the given monomial order.

1. \(f=4x^3 - 2xy + y^2,\ F=(x^2-y,xy-y^3)\) using lex.
2. \(f=3x^4-2xy^2+x^8,\ F=(x^3-3xy,y^2-2x)\) using grlex.

Find examples of each of the following. Justify your answers.

1. Two polynomials \(f,g \in \mathbb{R}[x]\) such that \(f \ne g\), but \(\langle f \rangle = \langle g \rangle\).
2. Two ideals \(I,J \subseteq \mathbb{R}[x]\) such that \(\mathbf{V}(I) = \mathbf{V}(J)\) but \(I \ne J\).
3. An ideal \(I \subseteq \mathbb{C}[x]\) such that \(\mathbf{V}(I) = \{0\}\) but \(I \ne \langle x \rangle\).
4. A polynomial \(f \in \mathbb{R}[x]\) such that \(\mathbf{V}(f)\) consists of exactly three distinct points.
5. A polynomial \(f \in \mathbb{R}[x]\) such that \(\mathbf{V}(f)\) consists of exactly three distinct points, but \(f\) has total degree \(4\).
6. An ideal \(I \subseteq \mathbb{R}[x,y,z]\) such that \(\mathbf{V}(I)\) is infinite.
7. An ideal \(I \subseteq \mathbb{R}[x,y,z]\) such that \(\mathbf{V}(I)\) is finite.
8. A sequence of polynomials \(f_1,\ldots,f_s\) in \(x,y\) such that \(\mathbf{V}(f_1, \ldots, f_s)\) is a non-empty finite set when the polynomials are considered as elements of \(\mathbb{R}[x,y]\), but \(\mathbf{V}(f_1, \ldots, f_s)\) is infinite when the polynomials are regarded as elements of \(\mathbb{C}[x,y]\).
9. A polynomial \(f\) in \(\mathbb{Q}[x,y,z]\) such that \(\operatorname{LT}(f)\) is different for each of the monomial orders lex, grlex, and grevlex.