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:
-
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.
-
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.
-
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.
-
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.
3 answers