Question

Which of the following shows that polynomials form a closed system under multiplication?

Option #1: 3(2x+5)=6x+15


Option #2: 8x+7=5x+3x+1+6


Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24


Option #4: 2x12⋅3=6x−−√

Answer 1

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.