A cereal box is 2 inches wide, 6.25 inches long, and 10 inches high. Another cereal box is x inches wide, 2x inches long, and 4x inches high. The total volume, in cubic inches, of the two cereal boxes can be represented by a polynomial P(x). Which of the following polynomials is a factor of P(x)?
25 - 10x + 4x² is a factor of P(x).
To find a polynomial that represents the total volume of the two cereal boxes, we first calculate the volume of each box. The first box has a volume of 2 × 6.25 × 10 = 125 cubic inches. The second box has a volume of x × 2x × 4x = 8x³ cubic inches. The total volume is then P(x) = 125 + 8x³, and the factorization reveals that 25 - 10x + 4x² is indeed a factor of this polynomial.
This polynomial does not factor into the total volume polynomial P(x). It represents a linear expression that does not align with the coefficients or structure of the cubic polynomial derived from the volumes of the cereal boxes.
Similar to the first choice, this polynomial is linear and does not correspond to the cubic nature of P(x). It cannot be factored out of a polynomial that contains a cubic term, making it an unsuitable choice.
While this expression resembles part of the polynomial P(x), it does not represent a factor. The presence of the constant term 125 indicates it does not encapsulate the polynomial's full structure necessary for being a factor.
This polynomial correctly factors into P(x). When we factor 8x³ + 125, we find that 25 - 10x + 4x² divides evenly, confirming it as a factor and demonstrating the relationship between the terms.
This polynomial, while similar to option D, has a different sign on the linear term and thus cannot be a factor of P(x). The signs and coefficients must precisely match those derived from the polynomial’s structure to be considered a valid factor.
The polynomial P(x) representing the total volume of the cereal boxes is 125 + 8x³. Among the options provided, 25 - 10x + 4x² stands out as a legitimate factor, reflecting the proper algebraic relationships derived from the volumes of both cereal boxes. The other choices either fail to meet the necessary criteria for factors or misrepresent the polynomial's structure.
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