Question
Which of the following responses demonstrates that polynomials form a closed system under multiplication?
A. x^2 + 2x + 1
B. (x^2 + 1) (x - 1/2)
C.(square root of x) (x + 1)
D.(x^2/2) (1/x)
A. x^2 + 2x + 1
B. (x^2 + 1) (x - 1/2)
C.(square root of x) (x + 1)
D.(x^2/2) (1/x)
Answers
Answered by
GPT-4o mini
To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to check if the product of the expressions is still a polynomial.
A polynomial is an expression made up of variables raised to non-negative integer powers and their coefficients.
Let's analyze each option:
A. \( x^2 + 2x + 1 \) - This is a polynomial on its own, but it does not demonstrate multiplication.
B. \( (x^2 + 1)(x - \frac{1}{2}) \) - This is the product of two polynomials. When multiplied, it will result in another polynomial. Therefore, it demonstrates closure under multiplication.
C. \( (\sqrt{x})(x + 1) \) - The square root of \( x \) is not a polynomial because it involves a fractional power (specifically, \( x^{1/2} \)). Thus, this product does not demonstrate closure.
D. \( \left( \frac{x^2}{2} \right) \left( \frac{1}{x} \right) \) - The product simplifies to \( \frac{x^2}{2x} = \frac{x}{2} \), which is a polynomial. However, one of the factors involved, \( \frac{1}{x} \), is not a polynomial (as it can be expressed as \( x^{-1} \)).
Thus, the only option that demonstrates that polynomials form a closed system under multiplication is:
**B. \( (x^2 + 1)(x - \frac{1}{2}) \)**.
A polynomial is an expression made up of variables raised to non-negative integer powers and their coefficients.
Let's analyze each option:
A. \( x^2 + 2x + 1 \) - This is a polynomial on its own, but it does not demonstrate multiplication.
B. \( (x^2 + 1)(x - \frac{1}{2}) \) - This is the product of two polynomials. When multiplied, it will result in another polynomial. Therefore, it demonstrates closure under multiplication.
C. \( (\sqrt{x})(x + 1) \) - The square root of \( x \) is not a polynomial because it involves a fractional power (specifically, \( x^{1/2} \)). Thus, this product does not demonstrate closure.
D. \( \left( \frac{x^2}{2} \right) \left( \frac{1}{x} \right) \) - The product simplifies to \( \frac{x^2}{2x} = \frac{x}{2} \), which is a polynomial. However, one of the factors involved, \( \frac{1}{x} \), is not a polynomial (as it can be expressed as \( x^{-1} \)).
Thus, the only option that demonstrates that polynomials form a closed system under multiplication is:
**B. \( (x^2 + 1)(x - \frac{1}{2}) \)**.