Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
(x−−√)(6x ⋅−3 )
Responses
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
(x−−√)(6x ⋅−3 )
Answers
Answered by
GPT-4o mini
To determine which of the responses shows that polynomials form a closed system under multiplication, we need to see if multiplying the terms results in a polynomial.
1. **14x³ ⋅ 5**: This simplifies to \( 70x^3 \), which is a polynomial.
2. **Start Fraction 1 over 4 x cubed End Fraction times 5**: This can be rewritten as \( \frac{5}{4} x^3 \), which is also a polynomial.
3. **3x²/4 - 1**: This is a rational expression and does not represent a multiplication of two polynomials.
4. **(x)(12x)**: This simplifies to \( 12x^2 \), which is a polynomial.
5. **(√x)(6x \cdot -3)**: This involves the square root of \( x \), which means it's not a polynomial.
From the analysis:
- The first choice, \( 14x^3 \cdot 5 \), is valid.
- The second choice, \( \frac{1}{4} x^3 \cdot 5 \) (simplifies to \( \frac{5}{4} x^3 \)), is valid.
- The fourth choice, \( (x)(12x) \) (simplifies to \( 12x^2 \)), is valid.
Thus, the valid responses that illustrate polynomials form a closed system under multiplication are:
1. \( 14x^3 \cdot 5 \)
2. \( \frac{1}{4} x^3 \cdot 5 \)
3. \( (x)(12x) \)
Since only one correct answer is required, you can select any of the first two from the valid responses, but likely the most straightforward one given is:
**14x³ ⋅ 5**, as it clearly shows multiplication of two polynomial terms resulting in another polynomial.
1. **14x³ ⋅ 5**: This simplifies to \( 70x^3 \), which is a polynomial.
2. **Start Fraction 1 over 4 x cubed End Fraction times 5**: This can be rewritten as \( \frac{5}{4} x^3 \), which is also a polynomial.
3. **3x²/4 - 1**: This is a rational expression and does not represent a multiplication of two polynomials.
4. **(x)(12x)**: This simplifies to \( 12x^2 \), which is a polynomial.
5. **(√x)(6x \cdot -3)**: This involves the square root of \( x \), which means it's not a polynomial.
From the analysis:
- The first choice, \( 14x^3 \cdot 5 \), is valid.
- The second choice, \( \frac{1}{4} x^3 \cdot 5 \) (simplifies to \( \frac{5}{4} x^3 \)), is valid.
- The fourth choice, \( (x)(12x) \) (simplifies to \( 12x^2 \)), is valid.
Thus, the valid responses that illustrate polynomials form a closed system under multiplication are:
1. \( 14x^3 \cdot 5 \)
2. \( \frac{1}{4} x^3 \cdot 5 \)
3. \( (x)(12x) \)
Since only one correct answer is required, you can select any of the first two from the valid responses, but likely the most straightforward one given is:
**14x³ ⋅ 5**, as it clearly shows multiplication of two polynomial terms resulting in another polynomial.