describing modeling with zero pairs\n\nif you are using algebra tiles to factor a trinomial of the form…

Explanation:

Step1: Identify the goal of algebra tiles

Algebra tiles are used to form a perfect rectangle where the area represents the trinomial $ax^2 + bx + c$.

Step2: Determine the condition for zero pairs

Zero pairs (one positive and one negative $x$-tile) are needed when the existing $x$-tiles cannot form a complete rectangle.

Step3: Explain the mathematical necessity

This occurs when the product $ac$ is negative or when the linear term $b$ does not provide enough tiles to fill the rectangular dimensions required by the $x^2$ and constant tiles.

Step4: Define the purpose of adding them

Adding zero pairs changes the appearance of the middle term without changing the total value, allowing the tiles to be rearranged into a rectangle.

Answer:

You would need to bring in zero pairs when the given tiles for $ax^2$, $bx$, and $c$ cannot be arranged into a complete rectangle. This typically happens when the constant term $c$ is negative or when additional $x$-tiles are required to fill the gaps in the rectangular array to make it factorable. Zero pairs allow you to add the necessary tiles to complete the shape without changing the overall value of the expression.