\begin{tabular}{l|c|}$$ \begin{array}{l} ext { 25a) Gary is building a square table for a kitchen. In his } \ ext { initial sketch, each side measured } x ext { feet. After rearranging } \ ext { some furniture, he realized he would have to add one foot }\end{array} $$ & $$ \begin{array}{c}25 \ ext { of } \ ext { to the length and remove one foot from the width and } \ ext { have a rectangular table instead. Write a polynomial to }\end{array} $$ \ \hline represent the area of the rectangular table. \end{tabular}

Question

\begin{tabular}{l|c|}$$ \begin{array}{l}	ext { 25a) Gary is building a square table for a kitchen. In his } \ 	ext { initial sketch, each side measured } x 	ext { feet. After rearranging } \ 	ext { some furniture, he realized he would have to add one foot }\end{array} $$ & $$ \begin{array}{c}25 \ 	ext { of } \ 	ext { to the length and remove one foot from the width and } \ 	ext { have a rectangular table instead. Write a polynomial to }\end{array} $$ \ \hline represent the area of the rectangular table. \end{tabular}

UpStudy AI · Accepted Answer

To determine the polynomial that represents the area of Gary's rectangular table, let's break down the problem step by step.

1. **Initial Table Dimensions:**
   - The table is originally a square with each side measuring $$ x $$ feet.
   - **Area of the square:** $$ \text{Area}_{\text{square}} = x \times x = x^2 $$ square feet.

2. **Modified Table Dimensions:**
   - **Length:** After rearranging, Gary adds 1 foot to the length. So, the new length is $$ x + 1 $$ feet.
   - **Width:** He removes 1 foot from the width. So, the new width is $$ x - 1 $$ feet.
   
3. **Area of the Rectangular Table:**
   - **Area of the rectangle:** $$ \text{Area}_{\text{rectangle}} = (\text{Length}) \times (\text{Width}) = (x + 1)(x - 1) $$.

4. **Expanding the Polynomial:**
   - Use the difference of squares formula: $$ (a + b)(a - b) = a^2 - b^2 $$.
   - Applying this to our expression: 
     $$
     (x + 1)(x - 1) = x^2 - 1
     $$

**Final Polynomial:**
$$
A(x) = x^2 - 1
$$

This polynomial $$ A(x) = x^2 - 1 $$ represents the area of the rectangular table after Gary rearranges the furniture.
The area of the rectangular table is $$ x^2 - 1 $$ square feet.

represent the area of the rectangular table.

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