To show that multiplying polynomials is a closed system, we will perform the multiplication of the polynomials \((x^2 - 1)\) and \((-2x + 5)\) and demonstrate that the result is also a polynomial.
Let's multiply the two polynomials step by step:
Step 1: Apply the Distributive Property (FOIL)
We will distribute each term in the first polynomial to each term in the second polynomial:
\[ (x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5 \]
Step 2: Calculate Each Term
Now, let’s compute each of these products:
- \(x^2 \cdot (-2x) = -2x^3\)
- \(x^2 \cdot 5 = 5x^2\)
- \((-1) \cdot (-2x) = 2x\)
- \((-1) \cdot 5 = -5\)
Step 3: Combine All the Terms
Now combine all these results:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Result
The multiplication of \((x^2 - 1)\) and \((-2x + 5)\) results in:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Conclusion
The result \(-2x^3 + 5x^2 + 2x - 5\) is clearly a polynomial.
This demonstrates that the system is closed under multiplication, as the product of two polynomials is itself a polynomial. Thus, multiplying polynomials does not produce an outcome outside the set of polynomials, confirming the closure property of polynomial multiplication.
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