Question

1. julio claims that you can find the area of a rectangle using the following method: take two positive integers x and y, where x > y. the side lengths of the rectangle are defined by the expressions 2x + y and 2x − y. the area of the rectangle is defined by the expression 4x² − y². is julio correct? explain your reasoning in the context of polynomial identities.

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the product of its length \( l \) and width \( w \), i.e., \( A = l \times w \).

Step2: Identify the length and width from the problem

Here, the length \( l = 2x + y \) and the width \( w = 2x - y \).

Step3: Calculate the product of length and width

We need to compute \( (2x + y)(2x - y) \). Using the difference of squares formula, which is a polynomial identity: \( (a + b)(a - b) = a^2 - b^2 \), where \( a = 2x \) and \( b = y \).
Substituting the values of \( a \) and \( b \) into the formula, we get:
\[

$$\begin{align*} (2x + y)(2x - y)&=(2x)^2 - y^2\\ &= 4x^2 - y^2 \end{align*}$$

\]
This is the same as the expression for the area that Julio defined.

Answer:

Julio is correct. By using the formula for the area of a rectangle (product of length and width) and the difference of squares polynomial identity \((a + b)(a - b)=a^2 - b^2\) (with \(a = 2x\) and \(b = y\)), the product of the side lengths \((2x + y)\) and \((2x - y)\) is \(4x^2 - y^2\), which matches the area expression Julio provided.