Question

Use the following multiplication problem to show how multiplying polynomials is a closed system. And answer the question

(xexponent2−1)(−2x+5)

Answer 1

To demonstrate that multiplying polynomials is a closed system, let's perform the multiplication of the polynomials \((x^2 - 1)(-2x + 5)\) and analyze the result.

### Step 1: Distribute
We'll use the distributive property (often called the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

1. Multiply \(x^2\) by \(-2x\):
\[
x^2 \cdot (-2x) = -2x^3
\]

2. Multiply \(x^2\) by \(5\):
\[
x^2 \cdot 5 = 5x^2
\]

3. Multiply \(-1\) by \(-2x\):
\[
-1 \cdot (-2x) = 2x
\]

4. Multiply \(-1\) by \(5\):
\[
-1 \cdot 5 = -5
\]

### Step 2: Combine All the Terms
Now we can combine all the results from the above multiplications:
\[
-2x^3 + 5x^2 + 2x - 5
\]

### Step 3: Final Result
Thus, the product of the polynomials \((x^2 - 1)(-2x + 5)\) is:
\[
-2x^3 + 5x^2 + 2x - 5
\]

### Analysis of Closure
A system is closed under an operation if applying the operation to members of the system produces a member that is also in the system. In this case:

- The system consists of polynomials.
- The product of two polynomials (in this case, \((x^2 - 1)\) and \((-2x + 5)\)) yields another polynomial \(-2x^3 + 5x^2 + 2x - 5\).

Since \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial (it consists of terms with non-negative integer exponents of \(x\) and real coefficients), we can conclude that the multiplication of these two polynomials is closed within the polynomial system.

**Conclusion:** The multiplication of polynomials is a closed system, as shown by the product \(-2x^3 + 5x^2 + 2x - 5\), which is also a polynomial.