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−−√


(1 point)

Answer 1

To show that polynomials form a closed system under multiplication, we need an example where multiplying two polynomials results in another polynomial.

Let's analyze the options:

- **Option #1:** \(3(2x+5)=6x+15\)
This is an example of distributing a number (3) with a polynomial, showing one polynomial operation, but it does not illustrate multiplication of two polynomials resulting in another polynomial.

- **Option #2:** \(8x+7=5x+3x+1+6\)
This is simply a rearrangement and combining like terms; it does not show multiplication.

- **Option #3:** \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\)
This is showing the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\), which results in \((6x^2y + 12x^2 - 12y - 24)\), a polynomial. This is an example that clearly shows that the product is also a polynomial.

- **Option #4:** \(2x^{12} \cdot 3=6x^{12}\)
This shows multiplication of a monomial and a constant, resulting in another monomial, which is a polynomial. However, it is a simpler case.

Among all the options, **Option #3** best illustrates that polynomials form a closed system under multiplication, as it involves the multiplication of two non-trivial polynomials leading to another polynomial.

Thus, the correct answer is **Option #3**.