Question
Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point)
Responses
(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis
(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis
x2+2x+1
x squared plus 2 x plus 1
(x2+1)(x−12)
Responses
(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis
(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis
x2+2x+1
x squared plus 2 x plus 1
(x2+1)(x−12)
Answers
Answered by
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply two polynomials together, the result is also a polynomial.
Let's analyze the provided responses:
1. \((\sqrt{x})(x + 1)\):
This expression includes \(\sqrt{x}\), which is not a polynomial. Thus, the product will not be a polynomial.
2. \(\left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right)\):
This expression simplifies to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is a polynomial.
3. \(x^2 + 2x + 1\):
This is already a polynomial, but it does not demonstrate multiplication of two polynomials.
4. \((x^2 + 1)(x - 12)\):
This expression is the product of two polynomials. When multiplied together, the result will also be a polynomial.
Among these options, the response \((x^2 + 1)(x - 12)\) clearly demonstrates multiplication of two polynomials, resulting in a polynomial.
So, the best option that demonstrates that polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - 12)\)**.
Let's analyze the provided responses:
1. \((\sqrt{x})(x + 1)\):
This expression includes \(\sqrt{x}\), which is not a polynomial. Thus, the product will not be a polynomial.
2. \(\left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right)\):
This expression simplifies to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is a polynomial.
3. \(x^2 + 2x + 1\):
This is already a polynomial, but it does not demonstrate multiplication of two polynomials.
4. \((x^2 + 1)(x - 12)\):
This expression is the product of two polynomials. When multiplied together, the result will also be a polynomial.
Among these options, the response \((x^2 + 1)(x - 12)\) clearly demonstrates multiplication of two polynomials, resulting in a polynomial.
So, the best option that demonstrates that polynomials form a closed system under multiplication is:
**\((x^2 + 1)(x - 12)\)**.