Question

modeling polynomials with algebra tiles

which geometric model using algebra tiles represents the factorization of (x^2 - 5x + 6)?

Explanation:

Factor the quadratic trinomial

Using the Factoring Trinomials knowledge point, we find two integers that multiply to \(+6\) and add up to \(-5\).

$$ x^2 - 5x + 6 = (x - 2)(x - 3) $$

Determine the dimensions of the algebra tile model

Using the Algebra Tiles knowledge point, the side lengths of the rectangle represent the factors.

$$ \text{Width} = x - 3,\quad \text{Height} = x - 2 $$

Match the factors to the visual models

We examine the headers along the top and left edges of each option:

• Option 1: Top header is \(x - 3\) (one \(+x\) tile and three \(-1\) tiles). Left header is \(x - 2\) (one \(+x\) tile and two \(-1\) tiles).
• Option 2: Top header is \(x - 2\). Left header is \(x - 2\).
• Option 3: Top header is \(x - 2\). Left header is \(x - 3\).
• Option 4: Top header is \(x - 3\). Left header is \(x - 3\).

Identify the correct geometric representation

Option 1 correctly represents the product of \((x - 3)\) along the top and \((x - 2)\) along the left:

• The top header has one \(+x\) tile and three \(-1\) tiles: \(x - 3\).
• The left header has one \(+x\) tile and two \(-1\) tiles: \(x - 2\).
• The inner area contains one \(x^2\) tile, five \(-x\) tiles, and six \(+1\) tiles, representing \(x^2 - 5x + 6\).

Answer:

• (A) Top header: \(x - 3\), Left header: \(x - 2\) (Correct answer)
• (B) Top header: \(x - 2\), Left header: \(x - 2\)
• (C) Top header: \(x - 2\), Left header: \(x - 3\)
• (D) Top header: \(x - 3\), Left header: \(x - 3\)