Question

Use the following multiplication problem to show how multiplying polynomials is a closed system. (x2−1)(−2x+5)

Answer 1

So the answer is?

Answer 2

What is the answer

Answer 3

To demonstrate that multiplying polynomials is a closed system, we will multiply the two given polynomials: \( (x^2 - 1)(-2x + 5) \).

1. **Identify the polynomials**:
- The first polynomial is \( x^2 - 1 \).
- The second polynomial is \( -2x + 5 \).

2. **Use the distributive property (also known as the FOIL method)** to multiply the polynomials:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 - 1 \cdot (-2x) - 1 \cdot 5
\]

3. **Calculate each term**:
- \( x^2 \cdot (-2x) = -2x^3 \)
- \( x^2 \cdot 5 = 5x^2 \)
- \( -1 \cdot (-2x) = 2x \)
- \( -1 \cdot 5 = -5 \)

4. **Combine all the terms**:
\[
-2x^3 + 5x^2 + 2x - 5
\]

Now let’s write the resulting polynomial clearly:
\[
-2x^3 + 5x^2 + 2x - 5
\]

5. **Conclusion**:
The result \( -2x^3 + 5x^2 + 2x - 5 \) is still a polynomial. This means that the product of two polynomials (of degrees 2 and 1) gives us another polynomial (of degree 3).

This demonstrates that the set of polynomials is closed under multiplication, as multiplying two polynomials results in another polynomial. The degree of the resulting polynomial (3) is less than or equal to the sum of the degrees of the factors (2 + 1 = 3), affirming that the multiplication of polynomials preserves the structure of polynomial expressions.