To determine if polynomials form a closed system under multiplication, we need to analyze the responses given to verify whether they result in polynomials.
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Response: \(\frac{3x^{2}}{4-1}\)
Simplifying this expression, we have: \[ \frac{3x^{2}}{4-1} = \frac{3x^{2}}{3} = x^{2} \] This is a polynomial.
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Response: \((x)(\frac{1}{2}x)\)
Multiplying these terms, we get: \[ (x)(\frac{1}{2}x) = \frac{1}{2}x^2 \] This is also a polynomial.
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Response: \((\sqrt{x})(6x\cdot -3)\)
First, simplify \(6x \cdot -3\): \[ 6x \cdot -3 = -18x \] Now multiplying \(\sqrt{x}\) and \(-18x\): \[ (\sqrt{x})(-18x) = -18\sqrt{x} \cdot x = -18x^{3/2} \] This is not a polynomial since \(\sqrt{x} = x^{1/2}\) has a fractional exponent.
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Response: \(\frac{1}{4x^{3}} \cdot 5\)
Simplifying this gives: \[ \frac{5}{4x^{3}} = \frac{5}{4}x^{-3} \] This is not a polynomial because it has a negative exponent.
From the above analysis, we see that only the first two responses yield polynomials. However, since the third and fourth responses result in non-polynomial expressions, we conclude that polynomials do not form a closed system under multiplication.
The responses show that:
- The first two are polynomials,
- The third and fourth are not.
Thus, the conclusion is that polynomials are not closed under multiplication based on the given options.
3 answers