Question

the trinomial \\(x^2 - 3x - 4\\) is represented by the model.

what are the factors of the trinomial?

(x + 1) and (x - 4)
(x + 4) and (x - 1)
(x + 5) and (x - 4)
(x + 4) and (x - 5)

Explanation:

Analyze the algebra tile model

The model shows an area representation of the trinomial \(x^2 - 3x - 4\).

• The top-left tile is \(+x^2\), which has dimensions \(x \times x\).
• The top-right tile is \(+x\), which has dimensions \(x \times 1\).
• The bottom-left column contains four \(-x\) tiles, which have dimensions \(x \times (-1)\) each, representing a total vertical length of \(-4\).
• The bottom-right column contains four \(-1\) tiles, which have dimensions \(1 \times (-1)\) each.

Determine the dimensions of the rectangle

The dimensions of the outer sides of the grid represent the factors:

• The horizontal dimension (top side) consists of one \(x\) tile width and one \(+1\) tile width: \(x + 1\).
• The vertical dimension (left side) consists of one \(x\) tile height and four \(-1\) tile heights: \(x - 4\).

Verify the product of the factors

Multiply the two binomial factors to confirm they equal the original trinomial:

$$ (x + 1)(x - 4) = x^2 - 4x + x - 4 = x^2 - 3x - 4 $$

This matches the given trinomial.

Answer:

• (A) \((x + 1)\) and \((x - 4)\) (Correct answer)
• (B) \((x + 4)\) and \((x - 1)\)
• (C) \((x + 5)\) and \((x - 4)\)
• (D) \((x + 4)\) and \((x - 5)\)