Question 11

D is a factor of P(x)

The polynomial P(x) can be expressed as the sum of the volumes of the two cereal boxes. The first box has a volume of 2 * 6.25 * 10 = 125 cubic inches. The second box, with dimensions x * 2x * 4x, has a volume of 8x³. Thus, P(x) = 125 + 8x³. The polynomial D) 25 - 10x + 4x² is a factor of P(x).

A) 5 - 2x

This polynomial does not factor into P(x). The structure of P(x) does not support a linear factor like 5 - 2x since it cannot be rearranged to produce the volume of either cereal box or their total volume.

B) 5 + 8x

This polynomial is also not a factor of P(x). While it includes the term 8x, it does not align with the necessary degrees or coefficients required to be a factor of the cubic polynomial formed by the volumes of the cereal boxes.

C) 125 + 8x

Although this polynomial shares part of the expression for P(x), it is not a factor. A factor must be able to divide P(x) completely without leaving a remainder, and this form does not fulfill that requirement.

D) 25 - 10x + 4x²

This polynomial is a factor of P(x). When factoring P(x) = 125 + 8x³, it can be shown through polynomial long division or synthetic division that D divides P(x) evenly, confirming it as a factor.

E) 25 + 10x + 4x²

This polynomial does not factor into P(x). Similar to option C, it does not align with the degree and structure of the polynomial formed by the volumes, thus cannot be considered a factor.

Conclusion

D) 25 - 10x + 4x² is definitively a factor of P(x), as it can divide the polynomial representing the total volume of the cereal boxes without leaving a remainder. The other options fail to meet the necessary criteria of being factors due to mismatched degrees or coefficients, confirming that D is the only correct choice.