To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to identify if the result of each multiplication operation yields a polynomial.
-
Response: ( 14x^3 \cdot 5 )
This results in \( 70x^3 \), which is a polynomial. -
Response: ( \sqrt{x} \cdot (6x \cdot -3) )
This equates to \( -18x^{3/2} \), which is not a polynomial because it involves a non-integer exponent. -
Response: ( x \cdot 2x )
This results in \( 2x^2 \), which is a polynomial. -
Response: ( 3x^{24} - 1 )
This is already a polynomial.
Based on the above evaluations, the responses that demonstrate multiplication resulting in a polynomial (thus showing closure under multiplication) are:
- \( 14x^3 \cdot 5 \)
- \( x \cdot 2x \)
Both of these responses confirm that polynomials form a closed system under multiplication.