To determine which of the responses shows that polynomials form a closed system under multiplication, we need to verify whether the result of the multiplication in each response is a polynomial. A polynomial is an expression made up of variables raised to non-negative integer powers and summed with coefficients.
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Response: \(2x^3 + x^2\)
This is a polynomial since it is a sum of terms where all variables (\(x\)) are raised to non-negative integer powers. -
Response: \(5x^{-1} \times 2\)
This expression contains \(x^{-1}\), which is a negative power of \(x\). Negative exponents do not satisfy the conditions of a polynomial. Therefore, this expression is not a polynomial. -
Response: \(\frac{1}{3} \left(\frac{x}{4} - 2\right)\)
This expression simplifies to \(\frac{x}{12} - \frac{2}{3}\), which is a polynomial because both terms have non-negative integer powers of \(x\) (the first term has \(x^1\) and the second term is a constant). -
Response: \(5 \times \frac{3}{x}\)
This expression contains \(\frac{3}{x}\) which can be rewritten as \(3x^{-1}\). Since \(x\) is raised to a negative exponent, this expression is not a polynomial.
From this analysis, the responses that demonstrate polynomials forming a closed system under multiplication are:
- \(2x^3 + x^2\)
- \(\frac{1}{3} \left(\frac{x}{4} - 2\right)\)
However, since only one response is needed regarding closure under multiplication, it can be stated that not all given responses demonstrate that polynomials form a closed system under multiplication, but \(2x^3 + x^2\) is a clear example of a polynomial.
1 answer