Sure, let's break this down step-by-step.
### Part A: Expression for the Area of the Rectangle
To find the area of the rectangle, we need to multiply the expressions for the lengths of its sides.
The sides of the rectangle are given as:
[tex]\[
\text{Side 1} = 4x + 5
\][/tex]
[tex]\[
\text{Side 2} = 3x + 10
\][/tex]
The area [tex]\( A \)[/tex] of a rectangle is given by the product of its length and width:
[tex]\[
A = (\text{Side 1}) \times (\text{Side 2}) = (4x + 5)(3x + 10)
\][/tex]
To find [tex]\( A \)[/tex], we can use the distributive property (also known as the FOIL method for binomials):
[tex]\[
A = (4x + 5)(3x + 10)
\][/tex]
First, distribute [tex]\( 4x \)[/tex] across [tex]\( 3x + 10 \)[/tex]:
[tex]\[
4x \cdot 3x + 4x \cdot 10 = 12x^2 + 40x
\][/tex]
Next, distribute [tex]\( 5 \)[/tex] across [tex]\( 3x + 10 \)[/tex]:
[tex]\[
5 \cdot 3x + 5 \cdot 10 = 15x + 50
\][/tex]
Combine these results:
[tex]\[
A = 12x^2 + 40x + 15x + 50 = 12x^2 + 55x + 50
\][/tex]
So, the expression for the area of the rectangle is:
[tex]\[
A = 12x^2 + 55x + 50
\][/tex]
### Part B: Degree and Classification of the Expression
The degree of a polynomial is determined by the highest power of the variable in the expression.
In the expression [tex]\( 12x^2 + 55x + 50 \)[/tex]:
- The term [tex]\( 12x^2 \)[/tex] has a degree of 2.
- The term [tex]\( 55x \)[/tex] has a degree of 1.
- The term [tex]\( 50 \)[/tex] has a degree of 0.
Thus, the highest degree term is [tex]\( 12x^2 \)[/tex], which implies that the degree of the polynomial is 2.
The classification of a polynomial is based on its 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 the degree of our polynomial is 2, it is classified as a quadratic polynomial.
So, the degree of the expression is 2, and it is classified as a quadratic polynomial.
### Part C: Demonstration of the Closure Property for Polynomials
The closure property for polynomials states that the set of polynomials is closed under addition, subtraction, and multiplication. This means that when you add, subtract, or multiply two polynomials, the result is always another polynomial.
In Part A, we multiplied two polynomials:
[tex]\[
(4x + 5) \text{ and } (3x + 10)
\][/tex]
Both of these expressions are polynomials. The product of these two polynomials is:
[tex]\[
12x^2 + 55x + 50
\][/tex]
This resulting expression is also a polynomial. Therefore, by multiplying two polynomials and obtaining another polynomial, we have demonstrated the closure property of polynomials.
So, Part A demonstrates the closure property of polynomials by showing that the product of two polynomials is indeed a polynomial.
