Question

A rectangle has sides whose lengths in units are represented by polynomial as shown where X is a positive integer

Answer 1

It seems like you're looking for a way to express the lengths of the sides of a rectangle using polynomials, where \(X\) is a positive integer. The sides of a rectangle typically have lengths represented as \(a\) and \(b\), which could be defined as polynomials of \(X\).

For instance, let’s say the lengths of the sides of the rectangle are given by the polynomials:
- Length of one side: \( a(X) = X^2 + 3X + 2 \)
- Length of the other side: \( b(X) = 2X + 1 \)

The area \(A\) of the rectangle can then be expressed as the product of the two lengths:
\[
A(X) = a(X) \times b(X) = (X^2 + 3X + 2)(2X + 1)
\]

To find the area you would multiply the two polynomials. Let’s do that:

1. Distribute \(X^2 + 3X + 2\) over \(2X + 1\):

\[
A(X) = (X^2 + 3X + 2)(2X + 1)
\]

2. Using the distributive property:

\[
= X^2 \cdot 2X + X^2 \cdot 1 + 3X \cdot 2X + 3X \cdot 1 + 2 \cdot 2X + 2 \cdot 1
\]

3. Simplifying this gives:

\[
= 2X^3 + X^2 + 6X^2 + 3X + 4X + 2
\]
\[
= 2X^3 + (1 + 6)X^2 + (3 + 4)X + 2
\]
\[
= 2X^3 + 7X^2 + 7X + 2
\]

So, the area of the rectangle in terms of \(X\) is given by:

\[
A(X) = 2X^3 + 7X^2 + 7X + 2
\]

Let me know if you would like to explore different polynomial expressions or any other aspect of rectangles!