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: Analyze polynomial modeling with rectangle

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 \(a = mn\), \(b=mq+np\), \(c = pq\). For \(2x^{2}+3x + 3\), using the discriminant \(D=b^{2}-4ac\) (where \(a = 2\), \(b = 3\), \(c = 3\)), we have \(D=3^{2}-4\times2\times3=9 - 24=- 15<0\). Since it cannot be factored over the real numbers, it cannot be represented as a rectangle (whose area is length\(\times\)width, corresponding to factoring the polynomial).

Step2: Analyze modeling with square

A square model would require \(a = c\) (for a polynomial \(ax^{2}+bx + c\)) and it to be a perfect - square trinomial. A perfect - square trinomial has the form \((mx + n)^{2}=m^{2}x^{2}+2mnx + n^{2}\). Here \(a = 2\), \(c = 3\), \(a
eq c\), and it is not a perfect - square trinomial (also confirmed by discriminant).

Step3: Analyze modeling with all positive tiles

We can use positive tiles for each term \(2x^{2}\) (2 \(x\times x\) tiles), \(3x\) (3 \(x\times1\) tiles), and \(3\) (3 \(1\times1\) tiles).

Step4: Analyze the number of tiles

The number of tiles is \(2 + 3+3=8\) (an even number).

Answer:

It cannot be modeled with a rectangle.