Polynomial closure refers to the property that when you perform certain operations on polynomials (such as addition, subtraction, or multiplication), the result is still a polynomial. Let's analyze the options:

A) $(x^2 + 3x + 7)(x + 2) = x^3 + 5x^2 + 13x + 14$

• This is a multiplication of two polynomials. Multiplying two polynomials results in another polynomial, and the given result is also a polynomial. Therefore, A demonstrates polynomial closure.

B) $(x^2 + 4x + 1) + (x^2 + x + 7) = 2x^2 + 5x + 8$

• This is the addition of two polynomials. Adding polynomials results in another polynomial, and the given result is also a polynomial. Therefore, B demonstrates polynomial closure.

C) $\frac{(x^2 + 5x + 6)}{(x + 2)(x + 1)} = \frac{(x + 3)}{(x + 1)}$

• This is a rational expression (a fraction involving polynomials), which is not necessarily a polynomial. Rational functions are not polynomials because division of polynomials is not closed under polynomial operations. Therefore, C does not demonstrate polynomial closure.

D) $(7x^4 + 5x^2 + 9) - (8x^4 - 3x + 2) = -x^4 + 5x^2 + 3x + 7$

• This is the subtraction of two polynomials. Subtracting polynomials results in another polynomial, and the given result is also a polynomial. Therefore, D demonstrates polynomial closure.

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Final answer:

• A, B, and D demonstrate polynomial closure.

Do you want further details or have any questions?

Here are 5 related questions:

1. What is the difference between polynomial closure and closure under rational functions?
2. How do you verify if a given operation on polynomials results in a polynomial?
3. What other operations preserve polynomial closure besides addition, subtraction, and multiplication?
4. Can the division of two polynomials ever result in another polynomial? If so, under what conditions?
5. How does polynomial closure relate to other algebraic structures like fields or rings?

Tip: When simplifying polynomials, remember that polynomial closure means the set of polynomials remains consistent through certain operations—like addition and multiplication—without introducing non-polynomial elements.