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9. [tex]$(06.02 \text{ MC})$[/tex]

The following table shows the length and width of a rectangle:

[tex]\[
\begin{tabular}{|l|l|l|}
\hline & \text{Length} & \text{Width} \\
\hline \text{Rectangle } A & 4x+5 & 3x-2 \\
\hline
\end{tabular}
\][/tex]

Which expression is the result of the perimeter of rectangle [tex]$A$[/tex] and demonstrates the closure property? (1 point)

A. [tex]$2x+6$[/tex]; the answer is a polynomial

B. [tex]$2x+6$[/tex]; the answer may or may not be a polynomial

C. [tex]$14x+6$[/tex]; the answer may or may not be a polynomial

D. [tex]$14x+6$[/tex]; the answer is a polynomial



Answer :

GinnyAnswer
To find the perimeter of rectangle A, we start by using the formula for the perimeter of a rectangle, which is given by:

[tex]\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \][/tex]

For rectangle A, the length is [tex]\(4x + 5\)[/tex] and the width is [tex]\(3x - 2\)[/tex]. We substitute these expressions into the perimeter formula:

[tex]\[ \text{Perimeter} = 2 \times \left( (4x + 5) + (3x - 2) \right) \][/tex]

Next, we simplify the expression inside the parentheses:

[tex]\[ (4x + 5) + (3x - 2) = 4x + 3x + 5 - 2 \][/tex]

Combine like terms:

[tex]\[ 4x + 3x + 5 - 2 = 7x + 3 \][/tex]

So, our perimeter expression becomes:

[tex]\[ \text{Perimeter} = 2 \times (7x + 3) \][/tex]

Now, we distribute the 2 through the expression inside the parentheses:

[tex]\[ 2 \times (7x + 3) = 2 \times 7x + 2 \times 3 = 14x + 6 \][/tex]

Therefore, the simplified expression for the perimeter of rectangle A is:

[tex]\[ 14x + 6 \][/tex]

Finally, let's address the closure property. A polynomial remains a polynomial after addition, subtraction, and multiplication, which are the operations we used in our steps. Therefore, the expression [tex]\(14x + 6\)[/tex] is indeed a polynomial.

So, the correct answer is:

[tex]\[ 14x + 6; \text{ the answer is a polynomial} \][/tex]

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