To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of any two polynomials is also a polynomial.
Let's analyze the options provided:
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\( x^2 + 2x + 1 \): This is a polynomial by itself, but it does not demonstrate a multiplication of two polynomials.
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\( (x + 1) \): This is also a polynomial but does not show a product.
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\( (2x + 1)(x - 1) \): This is a product of two polynomials. When multiplied, it yields \( 2x^2 - 2x + x - 1 = 2x^2 - x - 1 \), which is also a polynomial.
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\( (2)(4) \): This involves multiplication of two constants, which are technically polynomials of degree 0. However, it does not demonstrate the multiplication of two non-constant polynomials.
The response that best demonstrates the closure of polynomials under multiplication is:
(2x + 1)(x - 1).
1 answer