Sure! Let's break it down step by step.
When we multiply two polynomials, we're using a closed system, which means that when we multiply them, we only get other polynomials as the result.
Let's look at the problem \((x^2 - 1)(-2x + 5)\):
Step 1: Distribute Each Term We will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
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Multiply \(x^2\) by \(-2x\): \[ x^2 \cdot (-2x) = -2x^3 \]
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Multiply \(x^2\) by \(5\): \[ x^2 \cdot 5 = 5x^2 \]
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Multiply \(-1\) by \(-2x\): \[ -1 \cdot (-2x) = 2x \]
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Multiply \(-1\) by \(5\): \[ -1 \cdot 5 = -5 \]
Step 2: Combine All the Results Now we put all the results together: \[ -2x^3 + 5x^2 + 2x - 5 \]
Step 3: Write the Final Polynomial So the final result of \((x^2 - 1)(-2x + 5)\) is: \[ -2x^3 + 5x^2 + 2x - 5 \]
Conclusion: Since \(-2x^3 + 5x^2 + 2x - 5\) is also a polynomial, this shows that when we multiply two polynomials, we always get another polynomial as the result. That's why we say multiplying polynomials is a "closed system!"
And that's it! Multiplying polynomials just gives us more polynomials!
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