To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial. Let's analyze each of the options provided:
Option #1: \( 3(2x+5) = 6x + 15 \)
This is a multiplication of a polynomial \( (2x + 5) \) by a constant \( 3 \). The result \( 6x + 15 \) is also a polynomial, but this does not explicitly show that the product of two polynomials is a polynomial.
Option #2: \( 8x + 7 = 5x + 3x + 1 + 6 \)
This equality shows the simplification of a polynomial, but it does not illustrate multiplication of two polynomials.
Option #3: \( (2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24 \)
This is an example of multiplying two polynomials \( (2x^2 - 4) \) and \( (3y + 6) \). The result is \( 6x^2y + 12x^2 - 12y - 24 \), which is also a polynomial. This option clearly shows that the product of two polynomials results in another polynomial.
Option #4: \( 2x^{12} \cdot 3 = 6x \)
This is a multiplication of a polynomial term \( 2x^{12} \) by a constant \( 3 \). The result \( 6x^{12} \) is a polynomial, but this doesn't show the multiplication of two distinct polynomials.
Based on this analysis, Option #3 is the only one that demonstrates that polynomials form a closed system under multiplication. The multiplication of \( (2x^2 - 4) \) and \( (3y + 6) \) produces another polynomial, fulfilling the requirement.
3 answers