Difficulty: Easy
Correct Answer: 64
Explanation:
Introduction / Context:
This question relies on recognizing a standard algebraic identity for the difference of cubes. The expression (x − 4)(x^2 + 4x + 16) matches the pattern for x^3 − 4^3, allowing us to equate it with x^3 − P and identify P.
Given Data / Assumptions:
Concept / Approach:
The identity for the difference of cubes is a^3 − b^3 = (a − b)(a^2 + ab + b^2). Here, a corresponds to x and b corresponds to 4. This lets us directly match the given product with x^3 − 4^3 and read off P as 4^3.
Step-by-Step Solution:
Step 1: Recall the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2).
Step 2: Compare (x − 4)(x^2 + 4x + 16) with (a − b)(a^2 + ab + b^2).
Step 3: Identify a = x and b = 4, so a^2 + ab + b^2 becomes x^2 + 4x + 16, which matches the second factor exactly.
Step 4: Therefore (x − 4)(x^2 + 4x + 16) = x^3 − 4^3.
Step 5: Compute 4^3 = 64, so the right side is x^3 − 64.
Step 6: The equation in the question says (x − 4)(x^2 + 4x + 16) = x^3 − P, so by comparison P = 64.
Verification / Alternative check:
We can also expand the product directly. Multiplying (x − 4)(x^2 + 4x + 16) gives x^3 + 4x^2 + 16x − 4x^2 − 16x − 64, which simplifies to x^3 − 64. Again we see that P must be 64 for the expression to match x^3 − P for all x.
Why Other Options Are Wrong:
Options A (27), B (8), and E (16) correspond to 3^3, 2^3, and 2^4, which are unrelated to the 4^3 pattern in this identity. Option D (0) would incorrectly claim that the product equals x^3 for all x, which is false because substituting x = 4 yields zero on the left but 64 on the right when P = 0.
Common Pitfalls:
Many students mistake the identity for sum of cubes or attempt to expand under exam pressure and make algebraic errors. It is safer and faster to recognize the pattern directly as a difference of cubes and match the parameters a and b carefully.
Final Answer:
The constant P is 64.
Discussion & Comments