Question
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses (x−−√)(6x ⋅−3 ) open paren square root of x close paren times open paren 6 x times negative 3 close paren 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 14x3⋅5
Answers
GPT-4o mini
To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to identify whether the product of the given expressions results in another polynomial.
Let’s break down the responses:
1. \((\sqrt{x})(6x \cdot -3)\)
- Here, \(\sqrt{x}\) is not a polynomial, as polynomials cannot have fractional exponents. Therefore, this does not demonstrate a closed system.
2. \(3x^{24} - 1\)
- This expression is already a polynomial. However, it does not represent a multiplication, so it does not illustrate closure under multiplication.
3. \((x)(12x)\)
- This is a product of two polynomials: \(x\) and \(12x\). The product is \(12x^2\), which is also a polynomial. This demonstrates that polynomials are closed under multiplication.
4. \(\left(\frac{1}{2}\right)x\)
- This is a linear polynomial. However, it’s just a single term and does not illustrate closure on multiplication.
5. \(14x^3 \cdot 5\)
- This is also a multiplication of polynomials (one polynomial \(14x^3\) and the constant \(5\)). The product is \(70x^3\), which is a polynomial as well. This also demonstrates closure under multiplication.
From these, the most clear examples of multiplication that demonstrate closure would be:
- \((x)(12x)\)
- \(14x^3 \cdot 5\)
Therefore, the best responses showing that polynomials form a closed system under multiplication are:
- \((x)(12x)\)
- \(14x^3 \cdot 5\)
Let’s break down the responses:
1. \((\sqrt{x})(6x \cdot -3)\)
- Here, \(\sqrt{x}\) is not a polynomial, as polynomials cannot have fractional exponents. Therefore, this does not demonstrate a closed system.
2. \(3x^{24} - 1\)
- This expression is already a polynomial. However, it does not represent a multiplication, so it does not illustrate closure under multiplication.
3. \((x)(12x)\)
- This is a product of two polynomials: \(x\) and \(12x\). The product is \(12x^2\), which is also a polynomial. This demonstrates that polynomials are closed under multiplication.
4. \(\left(\frac{1}{2}\right)x\)
- This is a linear polynomial. However, it’s just a single term and does not illustrate closure on multiplication.
5. \(14x^3 \cdot 5\)
- This is also a multiplication of polynomials (one polynomial \(14x^3\) and the constant \(5\)). The product is \(70x^3\), which is a polynomial as well. This also demonstrates closure under multiplication.
From these, the most clear examples of multiplication that demonstrate closure would be:
- \((x)(12x)\)
- \(14x^3 \cdot 5\)
Therefore, the best responses showing that polynomials form a closed system under multiplication are:
- \((x)(12x)\)
- \(14x^3 \cdot 5\)
what is the right one
GPT-4o mini
The most straightforward response that clearly demonstrates that polynomials form a closed system under multiplication is:
**\((x)(12x)\)**
This response shows the multiplication of two polynomials \(x\) and \(12x\), resulting in \(12x^2\), which is also a polynomial.
So, **\((x)(12x)\)** is the correct choice.
**\((x)(12x)\)**
This response shows the multiplication of two polynomials \(x\) and \(12x\), resulting in \(12x^2\), which is also a polynomial.
So, **\((x)(12x)\)** is the correct choice.