Question

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Explanation:

Identify the factors to multiply

The problem asks to multiply \(3x(x - 1)\) using algebra tiles by following three steps:

1. Represent the first factor, \(3x\), along the vertical axis (Factor 1).
2. Represent the second factor, \(x - 1\), along the horizontal axis (Factor 2).
3. Complete the grid area (Product) by dragging tiles to represent the product of these factors.

Represent the factors with tiles

Using Algebra Tiles:

• Factor 1 (vertical): \(3x\) is represented by 3 positive green/orange \(x\)-tiles stacked vertically.
• Factor 2 (horizontal): \(x - 1\) is represented by 1 positive \(x\)-tile and 1 negative unit tile (\(-1\)) placed horizontally.

Determine the product area

Using Polynomial Multiplication:

• Multiplying each vertical \(x\)-tile by the horizontal \(x\)-tile yields an \(x^2\)-tile:

$$ x \cdot x = x^2 $$

Since there are 3 vertical \(x\)-tiles and 1 horizontal \(x\)-tile, we place 3 positive \(x^2\)-tiles in the first column of the product area.

• Multiplying each vertical \(x\)-tile by the horizontal \(-1\)-tile yields a \(-x\)-tile:

$$ x \cdot (-1) = -x $$

Since there are 3 vertical \(x\)-tiles and 1 horizontal \(-1\)-tile, we place 3 negative \(x\)-tiles in the second column of the product area.

Write the final algebraic expression

Summing the tiles in the product area:

• 3 positive \(x^2\)-tiles: \(3x^2\)
• 3 negative \(x\)-tiles: \(-3x\)

This matches the algebraic expansion:

$$ 3x(x - 1) = 3x^2 - 3x $$

Answer:

To complete the model for the product \(3x(x - 1)\):

1. Factor 1 (Vertical Column): Place 3 positive \(x\)-tiles.
2. Factor 2 (Horizontal Row): Place 1 positive \(x\)-tile and 1 negative unit tile (\(-1\)).
3. Product Area:

• Place 3 positive \(x^2\)-tiles in the column under the horizontal \(x\)-tile.
• Place 3 negative \(x\)-tiles in the column under the horizontal \(-1\)-tile.

The resulting simplified product is:

$$3x^2 - 3x$$