If p = 99, what is the exact numerical value of the algebraic expression p(p² + 3p + 3)?

Introduction / Context:
This problem involves evaluating a polynomial expression at a specific integer value. It looks straightforward but is also related to algebraic identities used in shortcuts, especially when p is chosen close to 100. Recognising patterns can help in mental calculation and speed improvement for competitive exams.

Given Data / Assumptions:

• p = 99.
• We need to compute p(p² + 3p + 3).
• All operations are basic integer arithmetic.

Concept / Approach:
Possible approaches:

• Direct substitution: plug p = 99 into the expression and compute step by step.
• Use algebraic expansion: p(p² + 3p + 3) = p³ + 3p² + 3p and then substitute p = 99.
• Notice that the result is very close to 10^6 since 99 is close to 100, which helps in checking plausibility.

Step-by-Step Solution:
Step 1: Start with the expression E = p(p² + 3p + 3). Step 2: Substitute p = 99 into the expression. Step 3: First compute p²: 99² = 9801. Step 4: Compute 3p: 3 × 99 = 297. Step 5: Evaluate the bracket: p² + 3p + 3 = 9801 + 297 + 3 = 10101. Step 6: Now E = p × 10101 = 99 × 10101. Step 7: Compute 10101 × 100 = 1010100, then subtract one group of 10101 to get 1010100 − 10101 = 999999. Step 8: Therefore the exact value of the expression is 999999.
Verification / Alternative check:
Step 1: Expand algebraically: p(p² + 3p + 3) = p³ + 3p² + 3p. Step 2: Substitute p = 99: p³ = 99³ = (100 − 1)³ = 1000000 − 3(10000) + 3(100) − 1 = 1000000 − 30000 + 300 − 1 = 970299. Step 3: Compute 3p²: 3 × 9801 = 29403. Step 4: Compute 3p: 3 × 99 = 297. Step 5: Add these: 970299 + 29403 + 297 = 999999, which matches the earlier result.
Why Other Options Are Wrong:
Option 1000000 is slightly larger than the correct value; it might result from mistakenly assuming p³ is exactly 10^6 when p is 100. Option 1000001 is even larger and does not match any careful calculation. Option 999998 is one less than the correct result and could come from a subtraction error during multiplication. Option 1000002 is not supported by any exact arithmetic from the expression.
Common Pitfalls:
Students may miscalculate 99² or 99³ due to rushing, especially with large numbers. Another mistake is to approximate 99 as 100 without adjusting the expression correctly, which leads to rough rather than exact values. Carrying errors while subtracting or adding intermediate values can also change the final answer by 1 or 2 units.
Final Answer:
The value of p(p² + 3p + 3) for p = 99 is 999999.