Question

Use the multiplication problem (x^2 - 1) (-2x + 5) to show how multiplying polynomials is a closed system

Answer 1

To show how multiplying polynomials is a closed system, we'll take the polynomial multiplication problem given: \((x^2 - 1)(-2x + 5)\), and demonstrate that the product of these two polynomials is still a polynomial.

### Step 1: Distributing Each Term

We'll use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials:

1. **Multiply \(x^2\) by each term in \(-2x + 5\)**:
\[
x^2 \cdot (-2x) = -2x^3
\]
\[
x^2 \cdot 5 = 5x^2
\]

2. **Multiply \(-1\) by each term in \(-2x + 5\)**:
\[
-1 \cdot (-2x) = 2x
\]
\[
-1 \cdot 5 = -5
\]

### Step 2: Combine All Terms

Now we combine all the results from the distributive steps:
\[
-2x^3 + 5x^2 + 2x - 5
\]

### Step 3: Write the Result as a Polynomial

The result of the multiplication is:
\[
-2x^3 + 5x^2 + 2x - 5
\]

### Conclusion: Closed System

The product \(-2x^3 + 5x^2 + 2x - 5\) is still a polynomial because it is expressed in the standard form of a polynomial (which consists of terms with non-negative integer exponents and coefficients).

This demonstrates that the set of polynomials is closed under multiplication: when you multiply any two polynomials, the result is always another polynomial. Thus, the operation of multiplying polynomials remains within the same set, confirming that it's a closed system.