In a previous lesson, you learned to add and subtract polynomials by combining like terms using the Commutative Property of Addition and the Distributive Property. You rearranged the terms of each expression then combined coefficients of like terms. Did you notice anything special about your solutions? Were the solutions polynomials? If you add or subtract polynomials, is the result also a polynomial? Is it possible to add or subtract polynomials and not end up with a polynomial? In other words, do polynomials form a closed system under addition and subtraction? In algebra, a closed system under addition and subtraction means that if you have a group of numbers, you can add or subtract any two numbers from that group, and the answer will always be another number in that same group. So, you won't end up with a number that's outside of the group. For example, when you add or subtract integers, the result is always an integer. If you add 3 and −5 , you get −2 , which is still an integer. Similarly, if you subtract −5 from 3 , you get 8 , which is also an integer. This shows that the set of integers is closed under addition and subtraction. Likewise, when you add or subtract rational numbers (numbers that can be expressed as fractions), the result is always a rational number. If you add 14 and 35 , you get 1720 , which is still a rational number. Subtracting 35 from 14 gives you −720 , which is also a rational number. This demonstrates that the set of rational numbers is closed under addition and subtraction. While working on the next problem, consider the following: When two polynomials are added or subtracted, the resulting expression is still considered a polynomial, even if it simplifies to a single term, a constant, or zero. A polynomial is defined as an expression with one or more terms, in which each term consists of a variable raised to a nonnegative integer exponent multiplied by a coefficient. If the expression simplifies to a single term, it can be considered a polynomial with just one term, in which the variable part has an exponent of zero and the coefficient represents the value of the constant term. Similarly, if the expression simplifies to zero, it can be viewed as a polynomial with no nonzero terms, as it still meets the criteria of having terms with variables raised to nonnegative integer exponents multiplied by coefficients.

Question

Using this info, provide me the answer to this question:
Use the following multiplication problem to show how multiplying polynomials is a closed system. (x2−1)(−2x+5) (2 points)

Answer 1

To demonstrate that the multiplication of polynomials is a closed system, let's multiply the two given polynomials, \((x^2 - 1)\) and \((-2x + 5)\).

Step 1: Distribute Each Term

We will use the distributive property (also known as the FOIL method when dealing with binomials) to multiply the polynomials.

\[ (x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 - 1 \cdot (-2x) - 1 \cdot 5 \]

Step 2: Perform the Multiplication

Calculating each term separately:

1. \(x^2 \cdot (-2x) = -2x^3\)
2. \(x^2 \cdot 5 = 5x^2\)
3. \(-1 \cdot (-2x) = 2x\)
4. \(-1 \cdot 5 = -5\)

Step 3: Combine Like Terms

Now, let's combine all these results:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion

The result of the multiplication \((-2x^3 + 5x^2 + 2x - 5)\) is a polynomial because it consists of terms where each term includes a coefficient multiplied by a variable raised to a nonnegative integer exponent (in this case, the exponents are 3, 2, 1, and 0).

Verifying Closed System

Since we started with two polynomials and multiplied them, and the result is still a polynomial, we can conclude that the set of polynomials is closed under multiplication. Thus, multiplying polynomials results in another polynomial, demonstrating that polynomials indeed form a closed system under multiplication.