To demonstrate how multiplying polynomials is a closed system, we will multiply the given polynomials \((x^2 - 1)\) and \((-2 + 5)\) in two steps.
Step 1: Simplify the Second Polynomial
First, simplify the second polynomial: \[ -2 + 5 = 3 \] Now our expression becomes: \[ (x^2 - 1)(3) \]
Step 2: Distribute (Multiply)
Now, we distribute \(3\) across the terms in the first polynomial: \[ 3(x^2 - 1) = 3x^2 - 3 \]
Final Result
The result of the multiplication is: \[ 3x^2 - 3 \]
Conclusion
The final expression, \(3x^2 - 3\), is still a polynomial, demonstrating that the operation of multiplying polynomials (a polynomial of degree 2 with a constant) results in another polynomial. This shows that the system is closed under multiplication.