To determine which response demonstrates that polynomials form a closed system under multiplication, we need to verify whether the product of the expressions provided results in a polynomial.
A polynomial is an expression that can be written in the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_i \) are coefficients and \( n \) is a non-negative integer.
Let's analyze each option:
A. \( x^2 + 2x + 1 \) - This is clearly a polynomial.
B. \( (x^2/2) (1/x) \) - This simplifies to \( x^{2-1}/2 = x/2 \), which is a polynomial but demonstrates division by \( x \), thus not conforming strictly to the definition of polynomial closure.
C. \( (\sqrt{x})(x+1) \) - Here, \( \sqrt{x} \) is \( x^{1/2} \). The product results in \( x^{1/2}(x + 1) = x^{1/2 + 1} + x^{1/2} = x^{3/2} + x^{1/2} \), which is not a polynomial since it has a fractional exponent.
D. \( (x^2 + 1)(x - 1/2) \) - This is the product of two polynomials, which results in \( x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \), confirming that it is indeed a polynomial.
Among the given options, the correct choice that demonstrates that polynomials form a closed system under multiplication is:
D. (x^2 + 1) (x - 1/2)
1 answer