Question

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

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


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


Option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24


Option #4: 2x1/2⋅3=6x−−√


(1 point)

Option # ___

Answer 1

To determine which of the options demonstrates that polynomials form a closed system under multiplication, we need to find the option that correctly shows the multiplication of two polynomials resulting in a polynomial.

Let's examine each option:

1. **Option #1: 3(2/x + 5) = 6/x + 15**
- The expression \(2/x + 5\) is not a polynomial because of the term \(2/x\) (it contains a variable in the denominator). Multiplying it by 3 does not yield a polynomial. So this option does not show closure.

2. **Option #2: 8x + 7 = 5x + 3x + 1 + 6**
- This option is simply rewriting a polynomial (the left-hand side is equal to the right-hand side), but does not involve multiplication. Thus, it does not demonstrate multiplication.

3. **Option #3: (2x² − 4)(3y + 6) = 6x²y + 12x² − 12y − 24**
- Both \(2x² - 4\) and \(3y + 6\) are polynomials. When we multiply them, we must get another polynomial. The result \(6x²y + 12x² - 12y - 24\) is indeed a polynomial. This option shows closure under multiplication.

4. **Option #4: 2x^(1/2) ⋅ 3 = 6x^(1/2)**
- The term \(2x^(1/2)\) is not a polynomial because polynomials have non-negative integer exponents. Therefore, this option does not demonstrate closure under multiplication.

Given the analysis, the option that shows that polynomials form a closed system under multiplication is:

**Option #3.**