Question

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

(x^2 - 1) (-2x + 5)

Answer 1

To demonstrate that multiplying polynomials is a closed system, we'll compute the product of the two polynomials \( (x^2 - 1) \) and \( (-2x + 5) \) and show that the result is also a polynomial.

### Step 1: Apply the Distributive Property

We can use the distributive property (also known as the FOIL method for binomials) to multiply these two polynomials. We'll distribute each term in the first polynomial by each term in the second polynomial:

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

### Step 2: Perform the Multiplications

Now, let's calculate each term:

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

### Step 3: Combine the Results

Now, combine all the results from the multiplications:

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

### Step 4: Write the Final Result

The final product of \( (x^2 - 1)(-2x + 5) \) is:

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

### Conclusion

The result \( -2x^3 + 5x^2 + 2x - 5 \) is a polynomial, which confirms that the multiplication of two polynomials produces another polynomial. This characteristic demonstrates that the set of polynomials is closed under multiplication, meaning that the operation (multiplication in this case) does not produce results outside the set of polynomials.