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 to demonstrate that the product of two polynomials is also a polynomial.

Among the options provided, let's analyze each one for polynomial multiplication:

- **Option #1**: \(3(2x+5)=6x+15\)
- This shows the multiplication of a constant (3) and a polynomial (2x + 5). The result is indeed a polynomial. However, this doesn't demonstrate the multiplication of two polynomials.

- **Option #2**: \(8x+7=5x+3x+1+6\)
- This is an equation representing a polynomial simplification, but it does not show multiplication, so it doesn't demonstrate closure under multiplication.

- **Option #3**: \((2x^2-4)(3y+6)=6x^2y+12x^2−12y−24\)
- This shows the multiplication of two polynomials \((2x^2 - 4)\) and \((3y + 6)\). The resulting expression is also a polynomial, which demonstrates closure under multiplication.

- **Option #4**: \(2x^{12} \cdot 3=6x^{12}\)
- Similar to option #1, this is the multiplication of a constant (3) and a polynomial (\(2x^{12}\)), resulting in a polynomial. However, like option #1, it does not show the multiplication of two polynomials.

**Conclusion**: The option that demonstrates polynomials forming a closed system under multiplication is **Option #3**: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\).