- The remaining area is determined by subtracting the removed area from the total area, resulting in the expression \(9x^2 - 16\).
- Using the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), where \(a=3x\) and \(b=4\), the expression is factored as \((3x - 4)(3x + 4)\).
- Comparing this result with the provided options confirms the correct expression is \(\boxed{(3x - 4)(3x + 4)}\).
Explanation
- Analyzing the Area of the Paper
To find the area of the remaining paper shape, we start with the initial area and subtract the area of the removed corner.
The total area of the original square is \(9x^2\) square centimeters.
A square corner with an area of \(16\) square centimeters is removed from it.
Therefore, the remaining area can be represented by the expression:
\[ 9x^2 - 16 \]
- Factoring the Expression
The expression \(9x^2 - 16\) follows the pattern of a difference of squares. The general formula for the difference of two squares is:
\[ a^2 - b^2 = (a - b)(a + b) \]
In our expression, we can identify \(a^2\) and \(b^2\) as:
\(a^2 = 9x^2\), which implies \(a = 3x\)
\(b^2 = 16\), which implies \(b = 4\)
By substituting \(a = 3x\) and \(b = 4\) into the formula, we get:
\[ (3x)^2 - (4)^2 = (3x - 4)(3x + 4) \]
- Conclusion
We compare our result, \((3x - 4)(3x + 4)\), with the provided options:
- \((x - 7)(x - 9)\)
- \((3x - 2)(3x - 8)\)
- \((3x - 4)(3x + 4)\)
- \((9x - 1)(x + 16)\)
The expression \((3x - 4)(3x + 4)\) perfectly matches the third option. Thus, the area of the remaining paper shape is \(\boxed{(3x - 4)(3x + 4)}\).
Examples
The difference of squares factorization is a fundamental algebraic skill used in various real-world scenarios. For example, in architecture or construction, if you are planning to cut a specific shape from a square material, knowing the area of the remaining section helps in calculating material waste or ensuring you have enough stock left for other parts of a project. By expressing the area as \((3x-4)(3x+4)\) square centimeters, you can easily calculate the remaining material for any given value of \(x\), optimizing resource usage and material costs efficiently.