Algebra II help

Question

Multiply (x3 + 4x2 - 5x - 8)(2x2 + 3x + 2). Is this operation closed for polynomials? Why or why not?

Answer 1

= 2x⁵ + 3x⁴ + 8x⁴ + 2x³ + 12x³ - 10x³ + 8x² -15x² -16x² - 10x - 24x -16

 

= 2x⁵ + 11x⁴ + 4x³ - 23x² - 34x - 16

 

And, yes multiplication of polynomials is closed. The multiplication of polynomials is accomplished by multiplying each possible pair of monomials from the polynomials and summing the products.  Since monomials are closed under multiplication and since polynomials are defined as the summation of monomial elements, the sum of the individual monomial products is by definition a polynomial.

Answer 2

Yes, multiplication of a polynomial by another polynomial is always a polynomial, so multiplication of polynomials is 'closed.'