To demonstrate that multiplying polynomials is a closed system using the multiplication problem \((x^2 - 1)(-2x + 5)\), we need to perform the multiplication and show that the result is also a polynomial.
Step 1: Distribute the terms
We will use the distributive property (also known as the FOIL method for binomials) to multiply the two polynomials:
\[ (x^2 - 1)(-2x + 5) \]
Distributing \(x^2\) over \(-2x + 5\):
\[ x^2 \cdot -2x = -2x^3 \] \[ x^2 \cdot 5 = 5x^2 \]
Now distributing \(-1\) over \(-2x + 5\):
\[ -1 \cdot -2x = 2x \] \[ -1 \cdot 5 = -5 \]
Step 2: Combine the results
Now we combine all the results from the distributions:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Final Result
The final result of multiplying the two polynomials \( (x^2 - 1)(-2x + 5) \) is:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Conclusion
The result \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial, demonstrating that the multiplication of two polynomials results in another polynomial. This property is what makes the set of polynomials a closed system under multiplication. Thus, we can conclude that multiplying polynomials is indeed a closed system.
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