Okay, so let's look at the multiplication problem: (x² - 1)(-2x + 5). To show how multiplying polynomials is a closed system, I’ll break it down step by step.
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Distribute Each Term: We need to distribute every term in the first polynomial (x² - 1) to every term in the second polynomial (-2x + 5).
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Multiply x² by Each Term:
- First, multiply x² by -2x:
- x² * -2x = -2x³
- Now, multiply x² by 5:
- x² * 5 = 5x²
- First, multiply x² by -2x:
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Multiply -1 by Each Term:
- First, multiply -1 by -2x:
- -1 * -2x = 2x
- Now, multiply -1 by 5:
- -1 * 5 = -5
- First, multiply -1 by -2x:
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Combine All Results: Now we combine everything we got from the distributions:
- So, we have: -2x³ + 5x² + 2x - 5
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Final Result: The final expression we get is -2x³ + 5x² + 2x - 5. This is a polynomial.
Now, here’s why it’s a closed system: When we multiply two polynomials (like (x² - 1) and (-2x + 5)), the result is always another polynomial. This means we don’t get anything weird like a fraction or a radical, just another polynomial. So we can keep multiplying polynomials, and we’ll always end up with a polynomial. That shows that this system is closed!
And that’s how we do it! Easy peasy!
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