Sure! Let's go through the problems step-by-step.
## Part A: Expression Representing the Area of the Rectangle
To find the area of a rectangle, we multiply its length by its width.
In this case, the sides of the rectangle are given by [tex]\( (4x + 5) \)[/tex] units and [tex]\( (3x + 10) \)[/tex] units.
The area [tex]\( A \)[/tex] of the rectangle can be expressed as:
[tex]\[ A = (4x + 5)(3x + 10) \][/tex]
Expanding this product, we use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(4x + 5)(3x + 10) = 4x \cdot 3x + 4x \cdot 10 + 5 \cdot 3x + 5 \cdot 10
\][/tex]
Now, we perform the multiplications:
[tex]\[
4x \cdot 3x = 12x^2
\][/tex]
[tex]\[
4x \cdot 10 = 40x
\][/tex]
[tex]\[
5 \cdot 3x = 15x
\][/tex]
[tex]\[
5 \cdot 10 = 50
\][/tex]
Adding all these terms together, we get:
[tex]\[
A = 12x^2 + 40x + 15x + 50
\][/tex]
Combining the like terms ([tex]\(40x\)[/tex] and [tex]\(15x\)[/tex]):
[tex]\[
A = 12x^2 + 55x + 50
\][/tex]
So, the expression that represents the area of the rectangle is:
[tex]\[ \boxed{12x^2 + 55x + 50} \][/tex]
## Part B: Degree and Classification of the Expression
Degree:
The degree of a polynomial is the highest power of the variable in the polynomial. In the expression [tex]\( 12x^2 + 55x + 50 \)[/tex], the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex], which is degree 2.
Classification:
The expression [tex]\( 12x^2 + 55x + 50 \)[/tex] is a polynomial. Specifically, it is a quadratic polynomial because the highest power of the variable [tex]\( x \)[/tex] is 2.
So the answer for Part B is:
- Degree: 2
- Classification: Quadratic polynomial
## Part C: Closure Property for Polynomials
The closure property states that the set of all polynomials is closed under addition, subtraction, and multiplication. This means that if you add, subtract, or multiply any two polynomials, the result is always another polynomial.
In Part A, we multiplied two binomials [tex]\((4x + 5)\)[/tex] and [tex]\((3x + 10)\)[/tex]:
[tex]\[
(4x + 5) \quad \text{and} \quad (3x + 10)
\][/tex]
Both are polynomials. Their product resulted in [tex]\( 12x^2 + 55x + 50 \)[/tex], which is also a polynomial.
Therefore, Part A demonstrates the closure property for polynomials because the product of two polynomials (the binomials) is itself a polynomial (the quadratic expression).
