To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomials is also a polynomial.
For example, consider two polynomials:
\( P(x) = 2x^2 + 3x + 1 \)
\( Q(x) = x + 4 \)
Now, let's multiply them:
\[ P(x) \cdot Q(x) = (2x^2 + 3x + 1)(x + 4) \]
Using the distributive property (or the FOIL method for binomials), we multiply:
\[ = 2x^2 \cdot x + 2x^2 \cdot 4 + 3x \cdot x + 3x \cdot 4 + 1 \cdot x + 1 \cdot 4 \]
Calculating each term gives us:
\[ = 2x^3 + 8x^2 + 3x^2 + 12x + x + 4 \] \[ = 2x^3 + (8x^2 + 3x^2) + (12x + x) + 4 \] \[ = 2x^3 + 11x^2 + 13x + 4 \]
since each term (the coefficients and the powers of x) is combined appropriately while maintaining polynomial form.
The result, \( 2x^3 + 11x^2 + 13x + 4 \), is also a polynomial. Thus, this example demonstrates closure under multiplication, which can be generalized: The product of any two polynomials results in another polynomial.
Therefore, any assertion that states that multiplying two polynomials results in another polynomial shows that polynomials form a closed system under multiplication.
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