To determine which option shows that polynomials form a closed system under multiplication, we need to find a case where multiplying two polynomials yields another polynomial.
Let's analyze the options:
-
Option #1: 3(2x+5)=6x+15
This shows a polynomial multiplied by a constant (which is also a polynomial, but of degree 0). The result is a polynomial. However, it does not illustrate the multiplication of two polynomials of higher degree. -
Option #2: 8x+7=5x+3x+1+6
This shows a sum of polynomials but does not demonstrate multiplication. It doesn’t address polynomial multiplication. -
Option #3: (2x²−4)(3y+6)=6x²y+12x²−12y−24
This option shows the multiplication of two polynomials: \(2x² - 4\) and \(3y + 6\), resulting in another polynomial. This is an example of polynomial multiplication producing another polynomial. -
Option #4: 2x¹²⋅3=6x
Here \(2x^{12}\) is a polynomial and 3 is a constant (also a polynomial of degree zero). The result is a polynomial, but like Option #1, it doesn't showcase the product of two non-constant polynomials.
Based on this analysis, Option #3 is the correct choice because it demonstrates that multiplying two polynomials yields another polynomial, proving that polynomials form a closed system under multiplication.