Problem: In this geometry problem, we are dealing with a square piece of paper with side length ( a ) cm. From this square, two rectangles with given dimensions are cut out, as shown in the image. The task is to determine which of the provided expressions is a factor of the polynomial that represents the area left after the rectangles are removed.
Solution:
-
Calculate the Area of the Square:
- The initial area of the square is ( a^2 ) square centimeters since it is a square with side length ( a ).
-
Dimensions of the Rectangles to be Removed:
- The yellow-shaded rectangles to be removed are labeled with dimensions ( (5b \times 2b) ) and ( (2b \times 3b) ).
-
Calculate the Area of Each Rectangle:
- Area of rectangle 1 (top left): ( 5b \times 2b = 10b^2 ).
- Area of rectangle 2 (bottom right): ( 2b \times 3b = 6b^2 ).
-
Total Area of Rectangles Removed:
- Add the areas of the two rectangles: ( 10b^2 + 6b^2 = 16b^2 ).
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Remaining Area of the Square:
- Subtract the total area of the rectangles from the area of the square:
[
\text{Remaining area} = a^2 - 16b^2
]
- Subtract the total area of the rectangles from the area of the square:
-
Factor the Polynomial:
- The expression ( a^2 - 16b^2 ) is a difference of squares, which can be factored as:
[
a^2 - 16b^2 = (a - 4b)(a + 4b)
]
- The expression ( a^2 - 16b^2 ) is a difference of squares, which can be factored as:
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Determine the Correct Option:
- From the factorization, the factors of the expression are ( (a - 4b) ) and ( (a + 4b) ).
- Given options are:
- A) ( a - 8b )
- B) ( a + 4 )
- C) ( a + 4b )
- D) ( a + 8b )
- The correct answer is option C) ( a + 4b ).
This comprehensive approach outlines the geometric reasoning and algebraic manipulation needed to solve the problem effectively.