Final answer:
Polynomials are closed under multiplication, meaning the product of any two polynomials is also a polynomial. This concept is essential in algebra and ensures that polynomials remain within the same mathematical set after multiplication.
Step-by-step explanation:
When we say that polynomials are closed under multiplication, it means that when you multiply two polynomials together, the result is also a polynomial. This is an important concept because it assures us that the set of polynomials is mathematically 'complete' in the sense that operating within it (with multiplication, in this case) does not lead us outside of it. It’s similar to saying that if you multiply two whole numbers together, you always get another whole number.
To understand this closure property, consider the example of multiplying (2x + 3) by (x - 1). Applying the FOIL method (First, Outer, Inner, Last), we get:
- First: 2x * x = 2x^2
- Outer: 2x * (-1) = -2x
- Inner: 3 * x = 3x
- Last: 3 * (-1) = -3
Putting it all together, we have 2x^2 - 2x + 3x - 3, which simplifies to 2x^2 + x - 3. This result is indeed a polynomial, illustrating the closure property of polynomials under multiplication.
This is crucial in algebra, as it allows for various operations, including factoring, solving polynomial equations, and working with polynomial functions without producing results that are 'non-polynomial,' which would require additional mathematical tools to handle.