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 given by $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l=(6x + 4)$ and $w=(2x + 11)$.

Step2: Multiply the polynomials

\[

$$\begin{align*} A&=(6x + 4)(2x+11)\\ &=6x\times(2x + 11)+4\times(2x + 11)\\ &=(6x\times2x+6x\times11)+(4\times2x + 4\times11)\\ &=12x^{2}+66x + 8x+44\\ &=12x^{2}+74x + 44 \end{align*}$$

\]

Step3: Determine degree and classification

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

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 (first - degree polynomials or 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 the polynomial $12x^{2}+74x + 44$, demonstrating the closure property of polynomials under multiplication.