Asked by l
Which of the following responses shows that polynomials form a closed system under multiplication?
(1 point)
Responses
14x3 ⋅ 5
14x3 ⋅ 5
x −−√ ⋅ (6x ⋅ −3)
x −−√ ⋅ (6x ⋅ −3)
x ⋅ 2x
x ⋅ 2x
3x24−1
(1 point)
Responses
14x3 ⋅ 5
14x3 ⋅ 5
x −−√ ⋅ (6x ⋅ −3)
x −−√ ⋅ (6x ⋅ −3)
x ⋅ 2x
x ⋅ 2x
3x24−1
Answers
Answered by
GPT-4o mini
To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to identify if the result of each multiplication operation yields a polynomial.
1. **Response: \(
14x^3 \cdot 5 \)**
This results in \( 70x^3 \), which is a polynomial.
2. **Response: \(
\sqrt{x} \cdot (6x \cdot -3) \)**
This equates to \( -18x^{3/2} \), which is not a polynomial because it involves a non-integer exponent.
3. **Response: \(
x \cdot 2x \)**
This results in \( 2x^2 \), which is a polynomial.
4. **Response: \(
3x^{24} - 1 \)**
This is already a polynomial.
Based on the above evaluations, the responses that demonstrate multiplication resulting in a polynomial (thus showing closure under multiplication) are:
- **\( 14x^3 \cdot 5 \)**
- **\( x \cdot 2x \)**
Both of these responses confirm that polynomials form a closed system under multiplication.
1. **Response: \(
14x^3 \cdot 5 \)**
This results in \( 70x^3 \), which is a polynomial.
2. **Response: \(
\sqrt{x} \cdot (6x \cdot -3) \)**
This equates to \( -18x^{3/2} \), which is not a polynomial because it involves a non-integer exponent.
3. **Response: \(
x \cdot 2x \)**
This results in \( 2x^2 \), which is a polynomial.
4. **Response: \(
3x^{24} - 1 \)**
This is already a polynomial.
Based on the above evaluations, the responses that demonstrate multiplication resulting in a polynomial (thus showing closure under multiplication) are:
- **\( 14x^3 \cdot 5 \)**
- **\( x \cdot 2x \)**
Both of these responses confirm that polynomials form a closed system under multiplication.