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)

Answer 1

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.

Answer 2

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.

Answer 3

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.