Closure Property of Polynomial Multiplication

Consider two arbitrary polynomials P(x) and Q(x) written in standard form. P(x) &= a_nx^(d_1) + ⋯ + a_1x + a_0 Q(x) &= b_nx^(d_2) + ⋯ + b_1x + b_0 These two polynomials can be multiplied by using the Distributive Property. The Product of Powers Property can also be applied to simplify the resulting expression.

Since P(x) and Q(x) are polynomials, all the exponents are whole numbers. Furthermore, because the whole numbers are closed under addition, the exponents of the resulting expression are whole numbers. Then, the new expression can be rewritten as follows. P(x)Q(x) &= c_kx^D + ⋯ + c_1x + c_0 Consequently, the new expression is a polynomial. Therefore, the product of two polynomials produces a polynomial, which proves that the polynomials are closed under multiplication.