Question

which statement is true about the prime polynomial $2x^2 + 3x + 3$?

• 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 Factoring

A prime polynomial (in quadratic form \(ax^2 + bx + c\)) cannot be factored into two binomials with integer coefficients. For \(2x^2 + 3x + 3\), the discriminant is \(b^2 - 4ac = 3^2 - 4(2)(3)=9 - 24=-15\), which is negative, so it doesn't factor over real integers.

Step2: Analyze Rectangle/Square Modeling

Modeling a polynomial with a rectangle (or square) corresponds to factoring it as \((mx + n)(px + q)\), where the length and width are the binomials. A square would require \(m = p\) and \(n = q\) (perfect square trinomial), but this polynomial isn't a perfect square (and not factorable).

• "It cannot be modeled with a rectangle": Since it's prime (not factorable into two binomials), it can't be represented as a rectangle (length×width from factoring).
• "It can be modeled with a square": A square needs factoring into identical binomials, which isn't possible here.
• "It cannot be modeled with all positive tiles": The polynomial has all positive terms, and modeling with tiles (area) uses positive tiles for positive coefficients, so this is false.
• "It can only be modeled with an odd number of tiles": The number of tiles (terms when expanded) is 3 (even? No, 3 is odd, but the key is factoring. However, the first statement is more fundamental—prime polynomials can't be rectangles because they don't factor.

Answer:

It cannot be modeled with a rectangle.