Question

multiplying polynomials: practice

\\( (2x + 1)(x + 1) \\)
\\( x(x + 4) \\)
\\( 2(x + 1) \\)
\\( (x + 2)(x + 2) \\)

Explanation:

🆕 New Concept Discovered: Algebra Tiles
Visualizing polynomial multiplication using area models

Step 1: Analyze the side lengths of the large square tile

The large blue square in the top-left has an area of \(x^2\). This means its dimensions are:

$$\text{Width} = x, \quad \text{Height} = x$$

Step 2: Determine the total width of the model

To the right of the \(x^2\) tile, there are two vertical rectangular tiles, each with an area of \(x\).
Since their height matches the height of the \(x^2\) tile (which is \(x\)), their width must be \(1\) each:

$$\text{Total Width} = x + 1 + 1 = x + 2$$

Step 3: Determine the total height of the model

Below the \(x^2\) tile, there are two horizontal rectangular tiles, each with an area of \(x\).
Since their width matches the width of the \(x^2\) tile (which is \(x\)), their height must be \(1\) each:

$$\text{Total Height} = x + 1 + 1 = x + 2$$

Step 4: Write the area as a product of dimensions

The total area of the algebra tile model is represented by multiplying its total width by its total height:

$$\text{Area} = (x + 2)(x + 2)$$

Answer:

\((x + 2)(x + 2)\)