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Sagot :
Certainly, let’s go through this step by step.
### Part A: Calculating the Area of the Rectangle
To find the area of a rectangle, we multiply its length by its width. Given:
- Length, \( L = 2x + 7 \)
- Width, \( W = 5x + 9 \)
The expression for the area, \( A \), of the rectangle is:
[tex]\[ A = L \times W \][/tex]
[tex]\[ A = (2x + 7) \times (5x + 9) \][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[ \begin{aligned} A &= (2x + 7)(5x + 9) \\ &= (2x \cdot 5x) + (2x \cdot 9) + (7 \cdot 5x) + (7 \cdot 9) \\ &= 10x^2 + 18x + 35x + 63 \\ &= 10x^2 + 53x + 63 \end{aligned} \][/tex]
So, the expression for the area of the rectangle is:
[tex]\[ A = 10x^2 + 53x + 63 \][/tex]
### Part B: Degree and Classification of the Expression
The degree of a polynomial is the highest power of the variable \( x \) in the expression. In the area expression \( 10x^2 + 53x + 63 \):
- The highest power of \( x \) is 2.
Thus, the degree of the polynomial is 2.
The classification of polynomials is based on their degree:
- A polynomial of degree 0 is a constant polynomial.
- A polynomial of degree 1 is a linear polynomial.
- A polynomial of degree 2 is a quadratic polynomial.
- A polynomial of degree 3 is a cubic polynomial, and so on.
Since our polynomial has a degree of 2, it is classified as a quadratic polynomial.
### Part C: Demonstrating the Closure Property for Polynomials
The closure property states that the set of polynomials is closed under addition, subtraction, and multiplication. This means that, when you perform these operations on polynomials, the result is always another polynomial.
In Part A, we multiplied two linear polynomials:
- \( 2x + 7 \)
- \( 5x + 9 \)
The result was:
- \( 10x^2 + 53x + 63 \), which is also a polynomial (specifically, a quadratic polynomial).
This demonstrates the closure property under multiplication for polynomials, as the product of two polynomials results in another polynomial.
### Summary
- Part A: The expression for the area is \( 10x^2 + 53x + 63 \).
- Part B: The degree of the polynomial is 2, and it is classified as a quadratic polynomial.
- Part C: This problem illustrates the closure property for polynomials, showing that the product of linear polynomials is another polynomial.
### Part A: Calculating the Area of the Rectangle
To find the area of a rectangle, we multiply its length by its width. Given:
- Length, \( L = 2x + 7 \)
- Width, \( W = 5x + 9 \)
The expression for the area, \( A \), of the rectangle is:
[tex]\[ A = L \times W \][/tex]
[tex]\[ A = (2x + 7) \times (5x + 9) \][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[ \begin{aligned} A &= (2x + 7)(5x + 9) \\ &= (2x \cdot 5x) + (2x \cdot 9) + (7 \cdot 5x) + (7 \cdot 9) \\ &= 10x^2 + 18x + 35x + 63 \\ &= 10x^2 + 53x + 63 \end{aligned} \][/tex]
So, the expression for the area of the rectangle is:
[tex]\[ A = 10x^2 + 53x + 63 \][/tex]
### Part B: Degree and Classification of the Expression
The degree of a polynomial is the highest power of the variable \( x \) in the expression. In the area expression \( 10x^2 + 53x + 63 \):
- The highest power of \( x \) is 2.
Thus, the degree of the polynomial is 2.
The classification of polynomials is based on their degree:
- A polynomial of degree 0 is a constant polynomial.
- A polynomial of degree 1 is a linear polynomial.
- A polynomial of degree 2 is a quadratic polynomial.
- A polynomial of degree 3 is a cubic polynomial, and so on.
Since our polynomial has a degree of 2, it is classified as a quadratic polynomial.
### Part C: Demonstrating the Closure Property for Polynomials
The closure property states that the set of polynomials is closed under addition, subtraction, and multiplication. This means that, when you perform these operations on polynomials, the result is always another polynomial.
In Part A, we multiplied two linear polynomials:
- \( 2x + 7 \)
- \( 5x + 9 \)
The result was:
- \( 10x^2 + 53x + 63 \), which is also a polynomial (specifically, a quadratic polynomial).
This demonstrates the closure property under multiplication for polynomials, as the product of two polynomials results in another polynomial.
### Summary
- Part A: The expression for the area is \( 10x^2 + 53x + 63 \).
- Part B: The degree of the polynomial is 2, and it is classified as a quadratic polynomial.
- Part C: This problem illustrates the closure property for polynomials, showing that the product of linear polynomials is another polynomial.
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