Question

which statement is true about the prime polynomial 2x² + 3x + 3? it cannot be modeled with a rectangle. it can be modeled with a square. it cannot be modeled with all positive tiles. it can only be modeled with an odd number of tiles.

Explanation:

Step1: Recall prime polynomial concept

A prime polynomial cannot be factored over the set of integers. In terms of area - model (using tiles or rectangles), a polynomial \(ax^{2}+bx + c\) can be modeled as a rectangle if it can be factored into two binomials \((mx + p)(nx+q)\) where \(mn=a\), \(pq = c\) and \(mq+np=b\). Since \(2x^{2}+3x + 3\) is prime, it cannot be factored into two binomials over the integers. So, it cannot be modeled as a rectangle.

Step2: Analyze square - modeling condition

For a polynomial to be modeled as a square, it must be a perfect - square trinomial of the form \((ax + b)^{2}=a^{2}x^{2}+2abx + b^{2}\). The given polynomial \(2x^{2}+3x + 3\) is not a perfect - square trinomial.

Step3: Consider positive tiles

We can use positive tiles to represent the terms \(2x^{2}\), \(3x\) and \(3\) since all the coefficients are positive.

Step4: Consider number of tiles

There is no rule that a prime polynomial can only be modeled with an odd number of tiles.

Answer:

It cannot be modeled with a rectangle.