Use the following multiplication problem to show how multiplying polynomials is a closed system (x^2 - 1) (-2x + 5)

Answer 1

To demonstrate how multiplying polynomials forms a closed system, let's multiply the given polynomials: \((x^2 - 1)\) and \((-2x + 5)\). A closed system means that when we carry out the operations (in this case, multiplication), we remain within the set of polynomials.

Step 1: Distribute the terms

We will distribute each term in the first polynomial by each term in the second polynomial:

\[ (x^2 - 1)(-2x + 5) = x^2 \cdot -2x + x^2 \cdot 5 - 1 \cdot -2x - 1 \cdot 5 \]

Step 2: Calculate each part

1. \(x^2 \cdot -2x = -2x^3\)
2. \(x^2 \cdot 5 = 5x^2\)
3. \(-1 \cdot -2x = 2x\)
4. \(-1 \cdot 5 = -5\)

Step 3: Combine like terms

Now, we will combine these results:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Step 4: Result as a polynomial

The result \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial of degree 3. Observing this result, we see that it is still in the form of a polynomial. Thus, the multiplication of the two polynomials has resulted in another polynomial.

Conclusion

This demonstrated that the multiplication of the polynomials \((x^2 - 1)\) and \((-2x + 5)\) produces another polynomial, indicating that the set of polynomials is closed under multiplication. In other words, multiplying polynomials together does not produce any results outside the set of polynomials.

Answer 2

can you explain without steps?

Answer 3

Certainly! When we multiply the polynomials \((x^2 - 1)\) and \((-2x + 5)\), we find that the result is \(-2x^3 + 5x^2 + 2x - 5\). This expression is still a polynomial, which means we have not moved outside the realm of polynomials.

This illustrates the concept of a closed system: when we multiply two polynomials, the product remains a polynomial. Therefore, the set of polynomials is closed under multiplication, as the operation produces results that stay within the same set.

Answer 4

make it different?

Answer 5

Sure! When we take the polynomials \((x^2 - 1)\) and \((-2x + 5)\) and multiply them together, the outcome is \(-2x^3 + 5x^2 + 2x - 5\). Since this result is itself a polynomial, it shows that multiplying polynomials does not lead us outside of their group.

In essence, this reinforces the idea that the multiplication of polynomials forms a closed system: no matter how many polynomials you multiply together, you will always end up with another polynomial, confirming that this structure is self-contained.