Sure! Let's look at the multiplication of the polynomials \((x^2 - 1)(-2x + 5)\) and see how this shows that multiplying polynomials is a closed system.
First, let's understand what a closed system means. It means that when we perform an operation (like multiplication) on elements of a certain set (like polynomials), we always get another element that is still within that same set (still a polynomial).
Now, let's multiply the two polynomials step by step:
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Distribute \(-2x\) to both terms in \(x^2 - 1\):
- \(-2x \cdot x^2 = -2x^3\)
- \(-2x \cdot (-1) = 2x\)
So, when we multiply \(-2x\) by \(x^2 - 1\), we get:
\(-2x^3 + 2x\) -
Now, distribute \(5\) to both terms in \(x^2 - 1\):
- \(5 \cdot x^2 = 5x^2\)
- \(5 \cdot (-1) = -5\)
So, when we multiply \(5\) by \(x^2 - 1\), we get:
\(5x^2 - 5\) -
Now, we add together all the results from the two steps:
- From the first step, we have: \(-2x^3 + 2x\)
- From the second step, we have: \(5x^2 - 5\)
Putting it all together, we get: \(-2x^3 + 5x^2 + 2x - 5\)
Now let’s look at what we have: \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial. It has terms with powers of \(x\), and it stays in the polynomial set.
Since we started with two polynomials and ended up with another polynomial, we can say that multiplication of polynomials is a closed system! This means no matter how many times we multiply polynomials, we will always get another polynomial as a result.
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