Let's find the expression for the perimeter of rectangle [tex]\( A \)[/tex] and demonstrate its characteristics.
To determine the perimeter of a rectangle, we use the formula:
[tex]\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \][/tex]
For rectangle [tex]\( A \)[/tex], the given lengths are:
- Length: [tex]\( 3x + 5 \)[/tex]
- Width: [tex]\( 2x - 3 \)[/tex]
First, add the length and width:
[tex]\[
\text{Length} + \text{Width} = (3x + 5) + (2x - 3)
\][/tex]
Combine like terms:
[tex]\[
\text{Length} + \text{Width} = 3x + 2x + 5 - 3 = 5x + 2
\][/tex]
Now, multiply by 2 to find the perimeter:
[tex]\[
\text{Perimeter} = 2 \times (5x + 2) = 10x + 4
\][/tex]
Thus, the expression for the perimeter of rectangle [tex]\( A \)[/tex] is:
[tex]\[ 10x + 4 \][/tex]
Let's analyze the closure property and the characteristics of the expression. The closure property refers to the set being closed under a specific operation. In this case, addition and multiplication of polynomials result in another polynomial.
The expression [tex]\( 10x + 4 \)[/tex] is a polynomial because:
1. It is a sum of terms of the form [tex]\( ax^n \)[/tex] where [tex]\( a \)[/tex] is a coefficient and [tex]\( n \)[/tex] is a non-negative integer.
2. The operation of addition and multiplication performed here did not produce any terms that fall outside the definition of a polynomial.
Therefore, the expression [tex]\( 10x + 4 \)[/tex] satisfies the closure property and is indeed a polynomial.
The correct answer is:
[tex]\[ 10x + 4; \text{ the answer is a polynomial} \][/tex]
