Use the following multiplication problem to show how multiplying polynomials is a closed system.

Question

(x^2 − 1) * (−2x + 5)

Answer 1

To show that multiplying polynomials is a closed system, we need to demonstrate that the product of two polynomials will also result in a polynomial.

Let's multiply the two given polynomials:

(x^2 − 1) * (−2x + 5)

Step 1: Distribute the first term of the first polynomial to each term in the second polynomial.
(x^2 * -2x) + (x^2 * 5) + (-1 * -2x) + (-1 * 5)

Step 2: Multiply the coefficients and add the exponents of x for each term.
-2x^3 + 5x^2 + 2x - 5

The result is a polynomial: -2x^3 + 5x^2 + 2x - 5.

Since the product of two polynomials, (x^2 − 1) and (−2x + 5), is also a polynomial, we have shown that multiplying polynomials is a closed system.