Sure! Let's look at the multiplication problem \((x^2 - 1)(-2x + 5)\) and see how multiplying polynomials works in a closed system.
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Write the Expression:
We start with \((x^2 - 1)(-2x + 5)\). -
Use the Distributive Property:
We will distribute each term in the first polynomial \((x^2 - 1)\) to each term in the second polynomial \((-2x + 5)\).-
First, distribute \(x^2\) to both terms in \((-2x + 5)\):
- \(x^2 \cdot -2x = -2x^3\)
- \(x^2 \cdot 5 = 5x^2\)
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Next, distribute \(-1\) to both terms in \((-2x + 5)\):
- \(-1 \cdot -2x = 2x\)
- \(-1 \cdot 5 = -5\)
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Combine All the Terms:
Now, we bring all those terms together: \(-2x^3 + 5x^2 + 2x - 5\) -
Final Result:
So, when we multiply, we get: \(-2x^3 + 5x^2 + 2x - 5\)
Conclusion:
This shows that when we multiply two polynomials, we still end up with another polynomial, which is why it's called a closed system. You can keep multiplying polynomials, and you won't get anything other than another polynomial!
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