### Part A: Expression for the Area of the Rectangle
To find the area of the rectangle, we need to multiply the expressions for its sides. Given the side lengths:
[tex]\[ \text{Side 1} = 2x + 7 \][/tex]
[tex]\[ \text{Side 2} = 5x + 9 \][/tex]
The area of a rectangle is given by the product of its length and width. Therefore, the area [tex]\(A\)[/tex] is:
[tex]\[ A = (2x + 7) \times (5x + 9) \][/tex]
To expand this expression, we use the distributive property (also known as the FOIL method for binomials):
First, multiply the first terms:
[tex]\[ 2x \cdot 5x = 10x^2 \][/tex]
Next, multiply the outer terms:
[tex]\[ 2x \cdot 9 = 18x \][/tex]
Then, multiply the inner terms:
[tex]\[ 7 \cdot 5x = 35x \][/tex]
Finally, multiply the last terms:
[tex]\[ 7 \cdot 9 = 63 \][/tex]
Now, add all these products together:
[tex]\[ A = 10x^2 + 18x + 35x + 63 \][/tex]
Combine like terms:
[tex]\[ A = 10x^2 + 53x + 63 \][/tex]
So, the expression that represents the area of the rectangle is:
[tex]\[ \boxed{10x^2 + 53x + 63} \][/tex]
### Part B: Degree and Classification of the Expression
The expression obtained for the area of the rectangle is:
[tex]\[ 10x^2 + 53x + 63 \][/tex]
To determine the degree of this polynomial, look at the term with the highest power of [tex]\(x\)[/tex]. Here, the highest power is [tex]\(2\)[/tex] (in the term [tex]\(10x^2\)[/tex]). Therefore, the degree of the polynomial is [tex]\(2\)[/tex].
Polynomials are classified based on their degree:
- A polynomial of degree 1 is called a linear polynomial.
- A polynomial of degree 2 is called a quadratic polynomial.
- A polynomial of degree 3 is called a cubic polynomial, and so on.
Since our polynomial has a degree of 2, it is classified as a quadratic polynomial.
Thus, the degree and classification of the expression [tex]\(10x^2 + 53x + 63\)[/tex] are:
[tex]\[ \text{Degree: } \boxed{2} \][/tex]
[tex]\[ \text{Classification: } \boxed{\text{Quadratic}} \][/tex]
### Part C: Demonstration of the Closure Property
The closure property for polynomials states that the sum, difference, or product of two polynomials is always a polynomial.
In Part A, we multiplied two binomials:
[tex]\[ (2x + 7) \text{ and } (5x + 9) \][/tex]
Each of these binomials is a polynomial. The result of their product, as expanded and simplified, is another polynomial:
[tex]\[ 10x^2 + 53x + 63 \][/tex]
This product is a polynomial itself, confirming that the set of polynomials is closed under multiplication. Therefore, Part A demonstrates the closure property for polynomials because it shows that multiplying two polynomials (in this case, binomials) results in another polynomial.
In conclusion, Part A demonstrates the closure property for polynomials by showing that the product of [tex]\((2x + 7)\)[/tex] and [tex]\((5x + 9)\)[/tex] is the polynomial [tex]\(10x^2 + 53x + 63\)[/tex]. Thus, polynomials are closed under multiplication.
