Which of the following responses demonstrates that polynomials form a closed system under multiplication?
A polynomial is an algebraic expression consisting of variables, coefficients, and exponentiation operators. In order to determine if polynomials form a closed system under multiplication, we need to determine if the product of polynomials is always another polynomial.
To demonstrate this, let's consider two arbitrary polynomials, P(x) and Q(x), where P(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0 and Q(x) = b_mx^m + b_(m-1)x^(m-1) + ... + b_1x + b_0.
We can multiply these two polynomials using the distributive property. Each term of P(x) will be multiplied by each term of Q(x), and then the like terms will be combined:
P(x) * Q(x) = (a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0) * (b_mx^m + b_(m-1)x^(m-1) + ... + b_1x + b_0)
= a_n*b_mx^(n+m) + (a_n*b_(m-1) + a_(n-1)*b_m)x^(n+m-1) + ... + (a_1*b_0 + a_0*b_1)x + a_0*b_0
As we can see, the product of polynomials is also a polynomial. Therefore, polynomials form a closed system under multiplication.
An example illustrating this closure property is:
Let P(x) = 2x^3 + 3x^2 + 4x + 1 and Q(x) = x^2 + 3x + 5.
P(x) * Q(x) = (2x^3 + 3x^2 + 4x + 1) * (x^2 + 3x + 5) = 2x^5 + 9x^4 + 24x^3 + 22x^2 + 23x + 5, which is also a polynomial.