To show that multiplying polynomials is a closed system, we can take the given polynomials \((x^2−1)\) and \((-2x+5)\) and perform the multiplication.
Step 1: Distribute the terms
We apply the distributive property (also known as the FOIL method for binomials).
\[ (x^2 - 1)(-2x + 5) \]
We can distribute each term in the first polynomial to each term in the second polynomial:
-
Multiply \(x^2\) by \(-2x\): \[ x^2 \cdot -2x = -2x^3 \]
-
Multiply \(x^2\) by \(5\): \[ x^2 \cdot 5 = 5x^2 \]
-
Multiply \(-1\) by \(-2x\): \[ -1 \cdot -2x = 2x \]
-
Multiply \(-1\) by \(5\): \[ -1 \cdot 5 = -5 \]
Step 2: Combine the results
Now, we combine all the terms together:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Step 3: Arrange the expression
We can arrange the final polynomial in standard form, which typically orders the terms from highest to lowest degree:
\[ -2x^3 + 5x^2 + 2x - 5 \]
Conclusion
The result, \(-2x^3 + 5x^2 + 2x - 5\), is also a polynomial. This shows that the product of two polynomials (both of degree 2 and 1, respectively) is still a polynomial (in this case, degree 3).
Therefore, the multiplication of polynomials is a closed system, as the product of two polynomials is also a polynomial.