Use multiplication to demonstrate these polynomials form a closed system.
(1/2 * x ^ 2 - 3)(4y ^ 3 + 5x ^ 2)
(1 point)
sqrt(x ^ 2) * y ^ 3 + 4x ^ 4 - 15x ^ 2
5/2 * x ^ 2 * y ^ 3 + 2x ^ 4 - 8y ^ 3 + 15x ^ 2
x ^ 2 * y ^ 3 + 5/(x ^ 4) - 8y ^ 3 - 15x ^ 2
2x ^ 2 * y ^ 3 + 5/2 * x ^ 4 - 12y ^ 3 - 15x ^ 2
To demonstrate that these polynomials form a closed system under multiplication, we need to show that the product of any two polynomials from the given set is also a polynomial from the given set.
Let's consider the first two polynomials:
(1/2 * x^2 - 3)(4y^3 + 5x^2) = (1/2 * x^2 * 4y^3) + (1/2 * x^2 * 5x^2) + (-3 * 4y^3) + (-3 * 5x^2)
= 2x^2y^3 + (5/2)x^4 - 12y^3 - 15x^2
Since the resulting polynomial is of the same form as the polynomials in the given set, it shows that these polynomials form a closed system under multiplication.
Now let's consider the product of the third and fourth polynomials:
(x^2 * y^3 + 5/(x ^ 4) - 8y ^ 3 - 15x ^ 2)(2x ^ 2 * y ^ 3 + 5/2 * x ^ 4 - 12y ^ 3 - 15x ^ 2) = (x^2 * y^3 * 2x^2 * y^3) + (x^2 * y^3 * 5/2 * x^4) + (x^2 * y^3 * -12y^3) + (x^2 * y^3 * -15x^2) + (5/(x^4) * 2x^2 * y^3) + (5/(x^4) * 5/2 * x^4) + (5/(x^4) * -12y^3) + (5/(x^4) * -15x^2) + (-8y^3 * 2x^2 * y^3) + (-8y^3 * 5/2 * x^4) + (-8y^3 * -12y^3) + (-8y^3 * -15x^2) + (-15x^2 * 2x^2 * y^3) + (-15x^2 * 5/2 * x^4) + (-15x^2 * -12y^3) + (-15x^2 * -15x^2)
= 2x^4y^6 + (5/2)x^6y^3 - 12x^2y^6 - 15x^4y^3 + 10y^3 + (25/2)x^2 - 60y^6 - 75x^2y^3 + 120x^2y^6 + (125/2)x^4 + 96y^6 + 225x^4 - 30x^2 + 90y^3 + (225/2)x^2
Again, we see that the resulting polynomial is of the same form as the polynomials in the given set, demonstrating that these polynomials form a closed system under multiplication.
Use multiplication to demonstrate these polynomials form a closed system.
(1/2 * x ^ 2 - 3)(4y ^ 3 + 5x ^ 2)
(1 point)
sqrt(x ^ 2) * y ^ 3 + 4x ^ 4 - 15x ^ 2
5/2 * x ^ 2 * y ^ 3 + 2x ^ 4 - 8y ^ 3 + 15x ^ 2
x ^ 2 * y ^ 3 + 5/(x ^ 4) - 8y ^ 3 - 15x ^ 2
2x ^ 2 * y ^ 3 + 5/2 * x ^ 4 - 12y ^ 3 - 15x ^ 2
To demonstrate that these polynomials form a closed system under multiplication, we need to show that the product of any two polynomials from the given set is also a polynomial from the given set.
Let's consider the first two polynomials:
(1/2 * x^2 - 3)(4y^3 + 5x^2) = (1/2 * x^2 * 4y^3) + (1/2 * x^2 * 5x^2) + (-3 * 4y^3) + (-3 * 5x^2)
= 2x^2y^3 + (5/2)x^4 - 12y^3 - 15x^2
Since the resulting polynomial is a combination of terms with the same variables (x and y) and the exponents are non-negative integers, it is also a polynomial from the given set.
Now let's consider the product of the third and fourth polynomials:
(x^2 * y^3 + 5/(x ^ 4) - 8y ^ 3 - 15x ^ 2)(2x ^ 2 * y ^ 3 + 5/2 * x ^ 4 - 12y ^ 3 - 15x ^ 2) = (x^2 * y^3 * 2x^2 * y^3) + (x^2 * y^3 * 5/2 * x^4) + (x^2 * y^3 * -12y^3) + (x^2 * y^3 * -15x^2) + (5/(x^4) * 2x^2 * y^3) + (5/(x^4) * 5/2 * x^4) + (5/(x^4) * -12y^3) + (5/(x^4) * -15x^2) + (-8y^3 * 2x^2 * y^3) + (-8y^3 * 5/2 * x^4) + (-8y^3 * -12y^3) + (-8y^3 * -15x^2) + (-15x^2 * 2x^2 * y^3) + (-15x^2 * 5/2 * x^4) + (-15x^2 * -12y^3) + (-15x^2 * -15x^2)
= 2x^4y^6 + (5/2)x^6y^3 - 12x^2y^6 - 15x^4y^3 + 10y^3 + (25/2)x^2 - 60y^6 - 75x^2y^3 + 120x^2y^6 + (125/2)x^4 + 96y^6 + 225x^4 - 30x^2 + 90y^3 + (225/2)x^2
The resulting polynomial is also a combination of terms with the same variables (x and y) and the exponents are non-negative integers, thus it is a polynomial from the given set.
Therefore, we have demonstrated that the given set of polynomials forms a closed system under multiplication.
To demonstrate that these polynomials form a closed system, we need to multiply them together and check for closure.
Let's multiply the given polynomials together:
(1/2 * x^2 - 3)(4y^3 + 5x^2)
Using the distributive property, we can multiply each term of the first polynomial by each term of the second polynomial:
1/2 * x^2 * 4y^3 + 1/2 * x^2 * 5x^2 - 3 * 4y^3 - 3 * 5x^2
Simplifying the multiplication:
2/2 * x^2 * y^3 + 5/2 * x^4 - 12y^3 - 15x^2
Reducing the fraction:
1 * x^2 * y^3 + 5/2 * x^4 - 12y^3 - 15x^2
Simplifying the constants:
x^2 * y^3 + 5/2 * x^4 - 12y^3 - 15x^2
The resulting polynomial is x^2 * y^3 + 5/2 * x^4 - 12y^3 - 15x^2.
Since the multiplication of the original polynomials resulted in another polynomial, we can conclude that these polynomials form a closed system.