### Part A: Factoring the Area of a Square
The area is given by the expression [tex]\( 16x^2 - 8x + 1 \)[/tex]. To find the side length of the square, we need to factor this expression completely.
1. Identify and rewrite the quadratic expression:
[tex]\[
16x^2 - 8x + 1
\][/tex]
2. Recognize it as a perfect square trinomial: Notice that the expression [tex]\( 16x^2 - 8x + 1 \)[/tex] fits the form [tex]\( (ax + b)^2 \)[/tex].
3. Check for factors:
[tex]\[
(4x - 1)^2 = (4x - 1)(4x - 1)
\][/tex]
Expanding [tex]\( (4x - 1)^2 \)[/tex] gives:
[tex]\[
(4x - 1)(4x - 1) = 16x^2 - 8x + 1
\][/tex]
Thus, the factored form of [tex]\( 16x^2 - 8x + 1 \)[/tex] is:
[tex]\[
(4x - 1)^2
\][/tex]
Therefore, the length of each side of the square is [tex]\( 4x - 1 \)[/tex].
### Part B: Factoring the Area of a Rectangle
The area is given by the expression [tex]\( 81x^2 - 4y^2 \)[/tex]. To find the dimensions of the rectangle, we need to factor this expression completely.
1. Identify and rewrite the expression:
[tex]\[
81x^2 - 4y^2
\][/tex]
2. Recognize it as a difference of squares: The expression [tex]\( 81x^2 - 4y^2 \)[/tex] fits the form [tex]\( a^2 - b^2 \)[/tex], which factors as [tex]\( (a - b)(a + b) \)[/tex].
3. Identify the squares:
[tex]\[
81x^2 = (9x)^2 \quad \text{and} \quad 4y^2 = (2y)^2
\][/tex]
4. Apply the difference of squares formula:
[tex]\[
81x^2 - 4y^2 = (9x - 2y)(9x + 2y)
\][/tex]
Therefore, the factored form of [tex]\( 81x^2 - 4y^2 \)[/tex] is:
[tex]\[
(9x - 2y)(9x + 2y)
\][/tex]
In conclusion, the dimensions of the rectangle are [tex]\( 9x - 2y \)[/tex] and [tex]\( 9x + 2y \)[/tex].
