answer:Bezout's theorem states that the number of intersection points of two plane curves with no common components is equal to the product of their degrees, counted with multiplicity. In this problem, we are given two plane curves C and D with degrees 3 and 4 respectively. Since they are not nested and have no common components, we can apply Bezout's theorem to find the number of intersection points. The number of intersection points is equal to the product of the degrees of the two curves: Number of intersection points = degree of C × degree of D Number of intersection points = 3 × 4 Number of intersection points = 12 So, the two plane curves C and D have 12 intersection points.

answer:To apply Bezout's theorem, we first need to find the degrees of the given algebraic curves. C1: x^2 + y^2 - 4 = 0 is a circle with degree 2 (since the highest power of x or y is 2). C2: x^2 - 2x + y^2 - 2y - 1 = 0 can be rewritten as (x - 1)^2 + (y - 1)^2 - 3 = 0, which is also a circle with degree 2. Bezout's theorem states that the number of intersection points of two algebraic curves is equal to the product of their degrees, counted with multiplicity, provided that the curves have no common components. Since both C1 and C2 are circles with degree 2 and have no common components, the number of intersection points is 2 * 2 = 4. Therefore, according to Bezout's theorem, there are 4 points of intersection between C1 and C2.

answer:Bezout's theorem states that the number of intersection points of two algebraic curves is equal to the product of their degrees, counted with multiplicity and including complex and points at infinity, provided that the curves have no common components. The given curves are: C1: x^2 + y^2 - 4 = 0 (degree 2) C2: x - 3y + 2 = 0 (degree 1) Since the two curves have no common components, we can apply Bezout's theorem: Number of intersection points = degree(C1) * degree(C2) = 2 * 1 = 2 So, there are 2 intersection points of these two curves, counted with multiplicity and including complex and points at infinity.

answer:Bezout's theorem states that the number of intersection points of two algebraic curves, counted with multiplicity, is equal to the product of the degrees of the curves, assuming that the curves have no common components. Let's first find the degrees of the given curves: C1: x^3 + y^3 - 3xy = 0 The degree of C1 is the highest power of any term, which is 3 (from x^3 and y^3). C2: x^2 + y^2 - 4x - 2y + 4 = 0 The degree of C2 is the highest power of any term, which is 2 (from x^2 and y^2). Now, we can apply Bezout's theorem: Number of intersection points = (degree of C1) * (degree of C2) = 3 * 2 = 6 So, there are 6 points of intersection of C1 and C2, counted with multiplicity.