Question

You used to call him with vacation problem to show how multiplying polynomials is a closed system ( x squared - 2)(-3x + 7)

Answer 1

To demonstrate that multiplying polynomials is a closed system using the example of \( (x^2 - 2)(-3x + 7) \), we will follow the steps of polynomial multiplication. The terms of each polynomial will be multiplied together, leading to a new polynomial.

### Step 1: Distribute Each Term

We will distribute each term from the first polynomial \( (x^2 - 2) \) to each term in the second polynomial \( (-3x + 7) \).

1. **Multiply \( x^2 \) with each term in \(-3x + 7\):**
\[
x^2 \cdot (-3x) = -3x^3
\]
\[
x^2 \cdot 7 = 7x^2
\]

2. **Multiply \(-2\) with each term in \(-3x + 7\):**
\[
-2 \cdot (-3x) = 6x
\]
\[
-2 \cdot 7 = -14
\]

### Step 2: Combine All the Results

Now, we will combine all the results from the multiplication:
\[
-3x^3 + 7x^2 + 6x - 14
\]

### Final Result

The result of multiplying the two polynomials \( (x^2 - 2)(-3x + 7) \) is:
\[
-3x^3 + 7x^2 + 6x - 14
\]

### Conclusion

The product \( -3x^3 + 7x^2 + 6x - 14 \) is itself a polynomial, showing that multiplying polynomials is indeed a closed system. In other words, the product of polynomials of degree \( m \) and \( n \) results in a polynomial of degree \( m+n \), maintaining the structure of polynomials.