Asked by r
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses 5x−1⋅2 5x−1⋅2 5 times x inverse times 2 2x3+x22x3+x2 2 x cubed plus x squared 5⋅3x5⋅3x 5 times 3 over x (13)(x4−2)
Answers
Answered by
r
What must be true about this multiplication problem? (9x3+76)(x−25)(9x3+76)(x−25) (1 point) Responses The result is a polynomial. The result is a polynomial. The result is not a closed system. The result is not a closed system. The result is not a polynomial. The result is not a polynomial. The result is unknown.
Answered by
GPT-4o mini
To show that polynomials form a closed system under multiplication, we need to demonstrate that when you multiply two polynomials together, the result is also a polynomial.
Looking at the provided options:
1. **\( 5x^{-1} \cdot 2 \)**: This expression involves \( x^{-1} \), which is not a polynomial (as polynomials only include non-negative integer powers of \( x \)). This result is not a polynomial.
2. **\( 2x^3 + x^2 \)**: This is a polynomial already, but it does not demonstrate a multiplication operation showing closure.
3. **\( 5 \cdot 3x \)**: This expression simplifies to \( 15x \), which is a polynomial.
4. **\( (13)(x^4 - 2) \)**: This involves multiplying a constant by a polynomial, which results in another polynomial \((13x^4 - 26)\).
Among the options listed, the second and fourth responses show polynomials. However, to demonstrate closure, the first response fails because it contains \( x^{-1} \).
The best option that illustrates closure under multiplication is **\( (13)(x^4 - 2) \)**, as it shows that multiplying a polynomial by a constant still yields a polynomial.
To summarize, polynomials are closed under multiplication because the product of any two polynomials results in another polynomial, and option 4 demonstrates this well.
Looking at the provided options:
1. **\( 5x^{-1} \cdot 2 \)**: This expression involves \( x^{-1} \), which is not a polynomial (as polynomials only include non-negative integer powers of \( x \)). This result is not a polynomial.
2. **\( 2x^3 + x^2 \)**: This is a polynomial already, but it does not demonstrate a multiplication operation showing closure.
3. **\( 5 \cdot 3x \)**: This expression simplifies to \( 15x \), which is a polynomial.
4. **\( (13)(x^4 - 2) \)**: This involves multiplying a constant by a polynomial, which results in another polynomial \((13x^4 - 26)\).
Among the options listed, the second and fourth responses show polynomials. However, to demonstrate closure, the first response fails because it contains \( x^{-1} \).
The best option that illustrates closure under multiplication is **\( (13)(x^4 - 2) \)**, as it shows that multiplying a polynomial by a constant still yields a polynomial.
To summarize, polynomials are closed under multiplication because the product of any two polynomials results in another polynomial, and option 4 demonstrates this well.
Answered by
GPT-4o mini
To evaluate the multiplication problem \((9x^3 + 76)(x - 25)\), we need to verify what can be inferred about the result of this multiplication.
1. **The Result is a Polynomial**: This statement is true. The expression consists of two polynomials being multiplied together. When two polynomials are multiplied, the result is also a polynomial. Specifically, \(9x^3 + 76\) is a polynomial of degree 3, and \(x - 25\) is a polynomial of degree 1. The resulting polynomial will be of degree \(3 + 1 = 4\).
2. **The Result is not a Closed System**: This statement is false in the context of polynomial operations. Polynomials form a closed system under addition, subtraction, and multiplication.
3. **The Result is not a Polynomial**: This statement is false. As stated earlier, the product of two polynomials is always a polynomial.
4. **The Result is Unknown**: This statement is misleading. While the specific form of the resulting polynomial may not be explicitly calculated here, we can confidently say that it is indeed a polynomial.
Given the above analysis, the correct response is:
**The result is a polynomial.**
1. **The Result is a Polynomial**: This statement is true. The expression consists of two polynomials being multiplied together. When two polynomials are multiplied, the result is also a polynomial. Specifically, \(9x^3 + 76\) is a polynomial of degree 3, and \(x - 25\) is a polynomial of degree 1. The resulting polynomial will be of degree \(3 + 1 = 4\).
2. **The Result is not a Closed System**: This statement is false in the context of polynomial operations. Polynomials form a closed system under addition, subtraction, and multiplication.
3. **The Result is not a Polynomial**: This statement is false. As stated earlier, the product of two polynomials is always a polynomial.
4. **The Result is Unknown**: This statement is misleading. While the specific form of the resulting polynomial may not be explicitly calculated here, we can confidently say that it is indeed a polynomial.
Given the above analysis, the correct response is:
**The result is a polynomial.**