To demonstrate that polynomials form a closed system under multiplication, we need to show that when two polynomials are multiplied, the result is also a polynomial.
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Identifying Polynomials: Let's take some example polynomials:
- \( P(x) = x^4 - 2 \)
- \( Q(x) = 2x^3 + x^2 \)
- \( R(x) = 5x + 2 \)
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Multiplication of Polynomials: If we take any two of the above polynomials and multiply them, we need to show that the result is still a polynomial.
Let's multiply \( P(x) \) and \( Q(x) \): \[ P(x) \cdot Q(x) = (x^4 - 2)(2x^3 + x^2) \]
Using the distributive property: \[ = x^4 \cdot 2x^3 + x^4 \cdot x^2 - 2 \cdot 2x^3 - 2 \cdot x^2 \] \[ = 2x^7 + x^6 - 4x^3 - 2x^2 \]
The result \( 2x^7 + x^6 - 4x^3 - 2x^2 \) is indeed a polynomial.
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Conclusion: Thus, since multiplying two polynomials \( P(x) \) and \( Q(x) \) resulted in another polynomial, we confirm that polynomials are closed under multiplication.
If you have a specific set of polynomial expressions or numbers to evaluate in mind (perhaps derived from your question), please clarify, and I'd be glad to assist further!
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