Answer:
The property of polynomial multiplication that states the product of two polynomials is always a polynomial is known as the "Closure Property" or "Closure under Multiplication."
In mathematics, the closure property means that when you perform a certain operation (in this case, multiplication) on elements within a certain set (in this case, the set of polynomials), the result of the operation also belongs to the same set.
For polynomials, this means that if you take any two polynomials, multiply them together, the result will always be another polynomial. In other words, polynomial multiplication is closed under the set of polynomials.
To illustrate with an example:
Let's take two polynomials:
1. P(x) = 2x^2 + 3x + 1
2. Q(x) = 4x + 5
Now, if we multiply these two polynomials, we get:
P(x) * Q(x) = (2x^2 + 3x + 1) * (4x + 5) = 8x^3 + 10x^2 + 12x + 5
As you can see, the product of the two polynomials (8x^3 + 10x^2 + 12x + 5) is also a polynomial, demonstrating the closure property of polynomial multiplication.
Explanation: