To determine if polynomials form a closed system under multiplication, we need to analyze if the product of two polynomials is also a polynomial.
- Response A: \( (x)(\frac{1}{2}x) \)
- Here, we have \( x \) (which is a polynomial of degree 1) and \( \frac{1}{2}x \) (which is also a polynomial of degree 1).
- The product is: \[ (x)(\frac{1}{2}x) = \frac{1}{2}x^2 \]
- \( \frac{1}{2}x^2 \) is a polynomial of degree 2.
Thus, Response A shows that the product of two polynomials is a polynomial.
- Response B: \( (\sqrt{x})(6x \cdot -3) \)
- Here, \( \sqrt{x} \) is not a polynomial (it can be expressed as \( x^{1/2} \), which is not a non-negative integer power).
- The term \( 6x \cdot -3 \) simplifies to \( -18x \), which is a polynomial of degree 1.
- The product is: \[ (\sqrt{x})(-18x) = -18x^{3/2} \]
- The term \( -18x^{3/2} \) is also not a polynomial because \( 3/2 \) is not an integer.
Thus, Response B does not show that polynomials form a closed system under multiplication since the multiplication involves a term that is not a polynomial.
Conclusion:
Response A is the correct choice as it demonstrates that polynomials form a closed system under multiplication.
1 answer