To determine whether polynomials form a closed system under multiplication, one must show that the product of any two polynomials is also a polynomial.
Definition of a Closed System
A set is said to be closed under an operation (in this case, multiplication) if performing that operation on members of the set always produces a member of the same set.
Polynomials
Polynomials are expressions composed of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. A general polynomial in one variable \( x \) can be expressed as:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]
where \( a_n, a_{n-1}, ..., a_0 \) are coefficients, and \( n \) is a non-negative integer.
Example of Polynomials under Multiplication
Let’s consider two polynomials:
- \( P(x) = 2x^2 + 3x + 1 \)
- \( Q(x) = x + 4 \)
Multiplying \( P(x) \) and \( Q(x) \):
\[ P(x) \cdot Q(x) = (2x^2 + 3x + 1)(x + 4) \] Using distribution: \[ = 2x^2 \cdot x + 2x^2 \cdot 4 + 3x \cdot x + 3x \cdot 4 + 1 \cdot x + 1 \cdot 4 \] \[ = 2x^3 + 8x^2 + 3x^2 + 12x + x + 4 \] Combining like terms: \[ = 2x^3 + 11x^2 + 13x + 4 \]
Result
The result \( 2x^3 + 11x^2 + 13x + 4 \) is again a polynomial.
Conclusion
Since the product of any two polynomials is a polynomial, we can conclude that the set of polynomials is indeed closed under multiplication.
Response
An appropriate response to illustrate this property might be: "Polynomials form a closed system under multiplication because the product of any two polynomials is always a polynomial as demonstrated by multiplying \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x + 4 \), which yields the polynomial \( 2x^3 + 11x^2 + 13x + 4 \)."
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