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
x2+2x+1
x squared plus 2 x plus 1
(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+1)(x−12)
Responses
(x−−√)(x+1)
left parenthesis Start Root x End Root right parenthesis left parenthesis x plus 1 right parenthesis
x2+2x+1
x squared plus 2 x plus 1
(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+1)(x−12)
Answers
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need a response that shows the result of multiplying two polynomials results in another polynomial.
Let's analyze the provided options:
1. **(√x)(x + 1)** - This expression involves the square root of x, which is not a polynomial, so this does not demonstrate closure in polynomials.
2. **x² + 2x + 1** - This is a polynomial, but it does not show multiplication involving two polynomials.
3. **(x²/2)(1/x)** - This expression simplifies to (x²/2) * (x^(-1)), which results in (1/2)x, a polynomial. However, since one of the factors is \(1/x\), which is not a polynomial, this does not demonstrate closure.
4. **(x² + 1)(x - 12)** - Both (x² + 1) and (x - 12) are polynomials. Their product, which would also be a polynomial, demonstrates closure under multiplication.
Since option 4, **(x² + 1)(x - 12)**, is the only response that involves multiplying two polynomials to yield another polynomial, it effectively demonstrates that polynomials form a closed system under multiplication.
**Correct Answer: (x² + 1)(x - 12)**
Let's analyze the provided options:
1. **(√x)(x + 1)** - This expression involves the square root of x, which is not a polynomial, so this does not demonstrate closure in polynomials.
2. **x² + 2x + 1** - This is a polynomial, but it does not show multiplication involving two polynomials.
3. **(x²/2)(1/x)** - This expression simplifies to (x²/2) * (x^(-1)), which results in (1/2)x, a polynomial. However, since one of the factors is \(1/x\), which is not a polynomial, this does not demonstrate closure.
4. **(x² + 1)(x - 12)** - Both (x² + 1) and (x - 12) are polynomials. Their product, which would also be a polynomial, demonstrates closure under multiplication.
Since option 4, **(x² + 1)(x - 12)**, is the only response that involves multiplying two polynomials to yield another polynomial, it effectively demonstrates that polynomials form a closed system under multiplication.
**Correct Answer: (x² + 1)(x - 12)**