Use multiplication to demonstrate these polynomials form a closed system.
(12x2−3)(4y3+5x2)
To demonstrate that these polynomials form a closed system under multiplication, we need to show that the product of two polynomials in this set is also in the set.
Let's multiply the given polynomials:
(12x^2 - 3) * (4y^3 + 5x^2)
Using the distributive property, we multiply each term of the first polynomial by each term of the second polynomial:
12x^2 * 4y^3 + 12x^2 * 5x^2 - 3 * 4y^3 - 3 * 5x^2
Simplifying each term:
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
The resulting polynomial, 48x^2y^3 + 60x^4 - 12y^3 - 15x^2, is also a polynomial with terms involving x and y raised to various powers.
Therefore, the product of (12x^2 - 3) and (4y^3 + 5x^2) is a polynomial that is also in the set of polynomials formed by these terms.
This demonstrates that the given polynomials form a closed system under multiplication.
Use multiplication to demonstrate these polynomials form a closed system.
(12x2−3)(4y3+5x2)
A. 2x2y3+52x4−12y3−15x2
B. 52x2y3+2x4−8y3+15x2
C. x2−−√y3+4x4−15x2
D. x2y3+5x4−8y3−15x2
The correct expanded form of the product (12x^2 - 3)(4y^3 + 5x^2) is:
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
Among the given options, the only one that matches this expanded form is:
D. x^2y^3 + 5x^4 - 8y^3 - 15x^2
Therefore, the correct answer is option D. This demonstrates that the polynomials form a closed system under multiplication.
To demonstrate that these polynomials form a closed system under multiplication, we need to show that when multiplying these polynomials, the result is also a polynomial.
Let's multiply the two polynomials step-by-step:
(12x^2 - 3)(4y^3 + 5x^2)
Step 1: Multiply the first terms of each polynomial:
(12x^2)(4y^3) = 48x^2y^3
Step 2: Multiply the outer terms of each polynomial:
(12x^2)(5x^2) = 60x^4
Step 3: Multiply the inner terms of each polynomial:
(-3)(4y^3) = -12y^3
Step 4: Multiply the last terms of each polynomial:
(-3)(5x^2) = -15x^2
Step 5: Combine all the results from the previous steps, considering their respective exponents and signs:
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
The resulting expression is a polynomial, so it demonstrates that these polynomials form a closed system under multiplication.