The Closure Property states that the product of two polynomials is also a polynomial. To demonstrate this using multiplication, let's take two simple polynomials and multiply them together.
Consider the following two polynomials:
- \( P(x) = 2x + 3 \)
- \( Q(x) = x^2 + 1 \)
Now we will multiply these two polynomials:
\[ P(x) \cdot Q(x) = (2x + 3)(x^2 + 1) \]
Using the distributive property (also known as the FOIL method in the case of binomials), we can calculate the product:
\[ = 2x \cdot x^2 + 2x \cdot 1 + 3 \cdot x^2 + 3 \cdot 1 \]
Calculating each term gives:
\[ = 2x^3 + 2x + 3x^2 + 3 \]
Now, let's rearrange the terms in standard form:
\[ = 2x^3 + 3x^2 + 2x + 3 \]
The result \( 2x^3 + 3x^2 + 2x + 3 \) is itself a polynomial.
Thus, this example demonstrates the Closure Property of multiplication of polynomials: multiplying two polynomials \( P(x) \) and \( Q(x) \) results in another polynomial.
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