Question

did cherise use algebra tiles to correctly represent the product of \\((x - 2)(x - 3)\\)?

no, she did not multiply the \\(x\\)-tiles by the negative integer tiles correctly.
no, she did not multiply the negative integer tiles by the other negative integer tiles correctly.
no, she did not add the terms together correctly.
yes, the product is \\(x^2 - 5x - 6\\).

Explanation:

Analyze the algebra tile representation

The factors to multiply are \((x - 2)\) and \((x - 3)\).

• The top row represents one factor: one \(+x\) tile (orange) and two \(-1\) tiles (blue). This correctly represents \((x - 2)\).
• The left column represents the other factor: one \(+x\) tile (orange) and three \(-1\) tiles (blue). This correctly represents \((x - 3)\).

Evaluate the multiplication inside the grid

• Multiplying \(+x\) by \(+x\) yields \(+x^2\) (orange), which is correct.
• Multiplying the top \(-1\) tiles by the left \(+x\) tile yields \(-x\) tiles (blue), which is correct.
• Multiplying the left \(-1\) tiles by the top \(+x\) tile yields \(-x\) tiles (blue), which is correct.
• Multiplying the left \(-1\) tiles by the top \(-1\) tiles should yield positive unit tiles (\(+1\)), because \((-1) \times (-1) = +1\).
• In the diagram, the unit tiles in the bottom-right are labeled \(-1\) (blue). This is incorrect because a negative times a negative must be positive.

Identify the correct option

• The error is in multiplying the negative integer tiles by the other negative integer tiles.
• Therefore, the correct option is: "No, she did not multiply the negative integer tiles by the other negative integer tiles correctly."

Answer:

• (A) No, she did not multiply the \(x\)-tiles by the negative integer tiles correctly.
• (B) No, she did not multiply the negative integer tiles by the other negative integer tiles correctly. (Correct answer)
• (C) No, she did not add the terms together correctly.
• (D) Yes, the product is \(x^2 - 5x - 6\).