Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x−−√)(6x ⋅−3 )
Responses
(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis
3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction
14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5
(x−−√)(6x ⋅−3 )
Answers
GPT-4o mini
To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we should look for products that ultimately result in a polynomial.
Let's evaluate each option:
1. **(x)(12x)**:
- The product is \( 12x^2 \), which is a polynomial. This shows closure under multiplication.
2. **(x)(1/2 x)**:
- The product is \( \frac{1}{2} x^2 \), which is also a polynomial. This demonstrates closure.
3. **(3x^2)/(4 - 1)**:
- This expression simplifies to \( \frac{3x^2}{3} = x^2 \), which is a polynomial. This illustrates closure, but it's a fraction form rather than a product.
4. **(1/4 x^3)(5)**:
- The product here is \( \frac{5}{4} x^3 \), which is still a polynomial, demonstrating closure.
5. **(√x)(6x \cdot -3)**:
- The expression involves a square root (not a polynomial) and hence does not exhibit closure since square roots do not yield a polynomial form.
Among these options, the first four display closure since their products result in polynomials. However, the option that most clearly states the property of closure under multiplication is the first one, **(x)(12x)**, as it is straightforward and clearly results in a typical polynomial form.
In conclusion, options 1, 2, 3, and 4 can demonstrate closure, but particularly, option 1 is the simplest representative of the concept.
Let's evaluate each option:
1. **(x)(12x)**:
- The product is \( 12x^2 \), which is a polynomial. This shows closure under multiplication.
2. **(x)(1/2 x)**:
- The product is \( \frac{1}{2} x^2 \), which is also a polynomial. This demonstrates closure.
3. **(3x^2)/(4 - 1)**:
- This expression simplifies to \( \frac{3x^2}{3} = x^2 \), which is a polynomial. This illustrates closure, but it's a fraction form rather than a product.
4. **(1/4 x^3)(5)**:
- The product here is \( \frac{5}{4} x^3 \), which is still a polynomial, demonstrating closure.
5. **(√x)(6x \cdot -3)**:
- The expression involves a square root (not a polynomial) and hence does not exhibit closure since square roots do not yield a polynomial form.
Among these options, the first four display closure since their products result in polynomials. However, the option that most clearly states the property of closure under multiplication is the first one, **(x)(12x)**, as it is straightforward and clearly results in a typical polynomial form.
In conclusion, options 1, 2, 3, and 4 can demonstrate closure, but particularly, option 1 is the simplest representative of the concept.