To demonstrate that multiplying polynomials is a closed system, we'll perform the multiplication of the given polynomials \( (x^2 - 1) \) and \( (-2x + 5) \).
We start by applying the distributive property, also known as the FOIL method in the case of binomials.
Here are the steps in detail:
- Distribute each term in the first polynomial to each term in the second polynomial:
\[ (x^2 - 1)(-2x + 5) = x^2(-2x) + x^2(5) - 1(-2x) - 1(5) \]
- Perform the multiplications:
- \( x^2(-2x) = -2x^3 \)
- \( x^2(5) = 5x^2 \)
- \( -1(-2x) = 2x \)
- \( -1(5) = -5 \)
- Combine all the results:
Now we can combine all the terms:
\[ -2x^3 + 5x^2 + 2x - 5 \]
- Write the final expression:
So, the product of \( (x^2 - 1)(-2x + 5) \) is:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Conclusion:
The resulting polynomial \( -2x^3 + 5x^2 + 2x - 5 \) is still a polynomial.
This demonstrates that when we multiply two polynomials, the result is also a polynomial, indicating that the set of polynomials is closed under multiplication.
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