To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial.
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Example Polynomials:
- \( P(x) = x^4 - 2 \)
- \( Q(x) = 2x^3 + x^2 \)
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Multiplication: 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 a polynomial.
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Conclusion: Thus, since the product of the polynomials \( P(x) \) and \( Q(x) \) is also a polynomial, we confirm that polynomials are closed under multiplication.