Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
A) 2x^3+x^2
2 x cubed plus x squared
B) 5x^−1⋅2
5 times x inverse times 2
C) 5⋅3/x
5 times 3 over x
D) (1/3)(x/4−2)
left parenthesis Start Fraction 1 over 3 End Fraction right parenthesis left parenthesis Start Fraction x over 4 End Fraction minus 2 right parenthesis
Responses
A) 2x^3+x^2
2 x cubed plus x squared
B) 5x^−1⋅2
5 times x inverse times 2
C) 5⋅3/x
5 times 3 over x
D) (1/3)(x/4−2)
left parenthesis Start Fraction 1 over 3 End Fraction right parenthesis left parenthesis Start Fraction x over 4 End Fraction minus 2 right parenthesis
Answers
Answer
The answer is D
Answered by
GPT-4o mini
To determine which response shows that polynomials form a closed system under multiplication, we need to recall that two polynomials can be multiplied together to produce another polynomial.
A polynomial is defined as an expression made up of variables raised to non-negative integer powers, and the coefficients can be any real numbers.
Let's analyze the provided options:
A) \(2x^3 + x^2\) - This is a polynomial.
B) \(5x^{-1} \cdot 2\) - This includes \(x^{-1}\), which is not a polynomial since it involves a negative exponent.
C) \( \frac{5 \cdot 3}{x} \) - This is also not a polynomial as it has \(x\) in the denominator (which is the same as having a negative exponent).
D) \(\frac{1}{3} \left( \frac{x}{4} - 2 \right)\) - This expression simplifies to a polynomial because it can be rewritten as \(\frac{1}{12} x - \frac{2}{3}\), which has no negative exponents.
From this breakdown, the only response that represents a polynomial is A), but it does not show the multiplication of two polynomials.
None of the given responses effectively demonstrates that the product of two polynomials remains a polynomial under multiplication.
However, to illustrate the closure property of polynomials under multiplication, a better example would be multiplying two polynomials directly rather than evaluating these particular expressions.
Based on the options you provided, A) is the only polynomial, but it does not exemplify multiplication that maintains closure in the polynomial system. Therefore, upon reviewing all the choices:
- A) is a polynomial but does not explicitly show closure under multiplication through an example.
- B) and C) definitely do not represent polynomials.
- D) simplifies to a polynomial but originates from a non-polynomial operation.
In conclusion, if looking strictly for polynomials, A) would be the valid choice, but please note it does not showcase the multiplication explicitly.
A polynomial is defined as an expression made up of variables raised to non-negative integer powers, and the coefficients can be any real numbers.
Let's analyze the provided options:
A) \(2x^3 + x^2\) - This is a polynomial.
B) \(5x^{-1} \cdot 2\) - This includes \(x^{-1}\), which is not a polynomial since it involves a negative exponent.
C) \( \frac{5 \cdot 3}{x} \) - This is also not a polynomial as it has \(x\) in the denominator (which is the same as having a negative exponent).
D) \(\frac{1}{3} \left( \frac{x}{4} - 2 \right)\) - This expression simplifies to a polynomial because it can be rewritten as \(\frac{1}{12} x - \frac{2}{3}\), which has no negative exponents.
From this breakdown, the only response that represents a polynomial is A), but it does not show the multiplication of two polynomials.
None of the given responses effectively demonstrates that the product of two polynomials remains a polynomial under multiplication.
However, to illustrate the closure property of polynomials under multiplication, a better example would be multiplying two polynomials directly rather than evaluating these particular expressions.
Based on the options you provided, A) is the only polynomial, but it does not exemplify multiplication that maintains closure in the polynomial system. Therefore, upon reviewing all the choices:
- A) is a polynomial but does not explicitly show closure under multiplication through an example.
- B) and C) definitely do not represent polynomials.
- D) simplifies to a polynomial but originates from a non-polynomial operation.
In conclusion, if looking strictly for polynomials, A) would be the valid choice, but please note it does not showcase the multiplication explicitly.
Answered by
GPT-4o mini
Thank you for your clarification. I believe there may have been a misunderstanding based on the details of option D.
Let's clearly reiterate the analysis of option D, which states:
**D) \(\frac{1}{3} \left( \frac{x}{4} - 2 \right)\)**
When simplified, this becomes:
\[
\frac{1}{3} \cdot \frac{x}{4} - \frac{1}{3} \cdot 2 = \frac{x}{12} - \frac{2}{3}
\]
This expression is a polynomial because it consists of a term with \(x\) raised to the first power and a constant term.
However, to confirm closure under multiplication as you mentioned, it is indeed important to confirm if the response illustrates a multiplication of polynomials:
- The expression \(\frac{x}{4} - 2\) represents a polynomial (as it is a sum of a term with \(x\) to a non-negative integer power and a constant).
- When multiplied by \(\frac{1}{3}\), it yields another polynomial.
Thus, choice D does show that when a polynomial is multiplied by a scalar (which is effectively what multiplying by \(\frac{1}{3}\) does), it remains a polynomial.
In light of your response, you are indeed correct that option D showcases how polynomials can interact and maintain their status as polynomials, thereby reflecting the closure property under multiplication. Thank you for your patience in clearing this up!
Let's clearly reiterate the analysis of option D, which states:
**D) \(\frac{1}{3} \left( \frac{x}{4} - 2 \right)\)**
When simplified, this becomes:
\[
\frac{1}{3} \cdot \frac{x}{4} - \frac{1}{3} \cdot 2 = \frac{x}{12} - \frac{2}{3}
\]
This expression is a polynomial because it consists of a term with \(x\) raised to the first power and a constant term.
However, to confirm closure under multiplication as you mentioned, it is indeed important to confirm if the response illustrates a multiplication of polynomials:
- The expression \(\frac{x}{4} - 2\) represents a polynomial (as it is a sum of a term with \(x\) to a non-negative integer power and a constant).
- When multiplied by \(\frac{1}{3}\), it yields another polynomial.
Thus, choice D does show that when a polynomial is multiplied by a scalar (which is effectively what multiplying by \(\frac{1}{3}\) does), it remains a polynomial.
In light of your response, you are indeed correct that option D showcases how polynomials can interact and maintain their status as polynomials, thereby reflecting the closure property under multiplication. Thank you for your patience in clearing this up!
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