Question

a rectangle has sides measuring (6x + 4) units and (2x + 11) units.
part a: what is the expression that represents the area of the rectangle? show your work. (4 points)
part b: what are the degree and classification of the expression obtained in part a? (3 points)
part c: how does part a demonstrate the closure property for polynomials? (3 points)

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l=(6x + 4)$ and $w=(2x + 11)$. So $A=(6x + 4)(2x+11)$.

Step2: Expand the product

Using the FOIL method:
\[

$$\begin{align*} A&=(6x\times2x)+(6x\times11)+(4\times2x)+(4\times11)\\ &=12x^{2}+66x + 8x+44\\ &=12x^{2}+74x + 44 \end{align*}$$

\]

Step3: Determine degree and classification for Part B

The degree of a polynomial is the highest - power of the variable. In the polynomial $12x^{2}+74x + 44$, the highest - power of $x$ is 2. A polynomial of degree 2 is called a quadratic polynomial.

Step4: Explain closure property for Part C

The closure property of polynomials under multiplication states that when we multiply two polynomials, the result is also a polynomial. In Part A, $(6x + 4)$ and $(2x + 11)$ are polynomials (linear polynomials), and their product $12x^{2}+74x + 44$ is also a polynomial (a quadratic polynomial).

Answer:

Part A: $12x^{2}+74x + 44$
Part B: Degree: 2, Classification: Quadratic polynomial
Part C: The product of two polynomials $(6x + 4)$ and $(2x + 11)$ is $12x^{2}+74x + 44$, which is also a polynomial, demonstrating the closure property of polynomials under multiplication.