Use algebra tiles to find \( \left(5 x^{2}-3 x+2\right)+\left(8 x^{2}+10\right) \)

To add the polynomials \( \left(5x^{2} - 3x + 2\right) \) and \( \left(8x^{2} + 10\right) \) using **algebra tiles**, follow these steps: ### **1. Understand Algebra Tiles** Algebra tiles are a visual representation tool used to model algebraic expressions. Each type of tile represents a different term: - **Large squares** represent \( x^{2} \) terms. - **Rectangles** represent \( x \) terms. - **Small squares** represent constant terms. Positive tiles are usually in one color (e.g., white) and negative tiles in another (e.g., black). ### **2. Represent Each Polynomial with Tiles** **First Polynomial: \( 5x^{2} - 3x + 2 \)** - **\( 5x^{2} \)**: Five large \( x^{2} \) tiles. - **\(-3x\)**: Three negative \( x \) tiles. - **\( +2 \)**: Two positive small squares. **Second Polynomial: \( 8x^{2} + 10 \)** - **\( 8x^{2} \)**: Eight large \( x^{2} \) tiles. - **\( +10 \)**: Ten positive small squares. - *(Note: There are no \( x \) terms in the second polynomial.)* ### **3. Combine Like Terms Using the Tiles** - **Combine \( x^{2} \) Terms:** - \( 5x^{2} + 8x^{2} = 13x^{2} \) - So, you now have thirteen large \( x^{2} \) tiles. - **Combine \( x \) Terms:** - Only the first polynomial has \( x \) terms: \(-3x\) - So, you have three negative \( x \) tiles. - **Combine Constant Terms:** - \( 2 + 10 = 12 \) - So, you have twelve positive small squares. ### **4. Write the Final Expression** After combining all like terms, the resulting polynomial is: \[ 13x^{2} - 3x + 12 \] ### **Visualization with Algebra Tiles** Here's a simplified representation: - **Large \( x^{2} \) Tiles:** 13 positive - **\( x \) Tiles:** 3 negative - **Small Squares:** 12 positive ``` [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ x² ] [ -x ] [ -x ] [ -x ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] [ + ] ``` This arrangement visually confirms that after combining like terms, the sum of the two polynomials is \( 13x^{2} - 3x + 12 \). **Final Answer:** \[ \left(5x^{2} - 3x + 2\right) + \left(8x^{2} + 10\right) = 13x^{2} - 3x + 12 \]