To demonstrate that multiplying polynomials is a closed system, we will multiply the two given polynomials: \((x^2 - 2)\) and \((-3x + 7)\). A closed system in this context means that the result of the multiplication will also be a polynomial.
Step 1: Use the distributive property
We will apply the distributive property (also known as the FOIL method for binomials) to multiply the two expressions. Each term in the first polynomial needs to be multiplied by each term in the second polynomial.
Let's break it down:
\[ (x^2 - 2)(-3x + 7) = x^2(-3x) + x^2(7) - 2(-3x) - 2(7) \]
Step 2: Perform the multiplications
Now, we multiply each of the terms:
- \(x^2(-3x) = -3x^3\)
- \(x^2(7) = 7x^2\)
- \(-2(-3x) = 6x\)
- \(-2(7) = -14\)
Step 3: Combine all the terms
Now we combine all the terms we found:
\[ -3x^3 + 7x^2 + 6x - 14 \]
Step 4: Write the final polynomial
The result of the multiplication is:
\[ -3x^3 + 7x^2 + 6x - 14 \]
Conclusion: Closed System
The product \(-3x^3 + 7x^2 + 6x - 14\) is also a polynomial because it is expressed as a sum of terms of the form \(ax^n\), where \(a\) is a coefficient (which can be any real number) and \(n\) is a non-negative integer.
Thus, we have shown that multiplying polynomials yields another polynomial, confirming that the operation of polynomial multiplication is a closed system.
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