Question

If $x = 1 + 2i$, then the value of $x^3 + 7x^2 - x + 16$ is 

• A. $-17 - 24i$
• B. $-17 + 24i$
• C. $17 - 24i$
• D. $17 + 24i$

Hint:

Constructing a quadratic equation equal to zero ($x^2 - 2x + 5 = 0$) is an exceptional algebraic shortcut. It completely bypasses the tedious and error-prone process of calculating complex cubics like $(1+2i)^3$, reducing the problem to basic mental math.
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Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Given x = 1 + 2i, evaluate the polynomial expression x³ + 7x² - x + 16.

Step 2: Key Formula or Approach:
Form a quadratic equation from x = 1 + 2i by isolating i and squaring: (x - 1)² = (2i)² → x² - 2x + 1 = -4 → x² - 2x + 5 = 0. Use this relation to repeatedly reduce the degree of the expression.

Step 3: Detailed Explanation:
Since x² - 2x + 5 = 0, rewrite the polynomial by extracting this zero-valued quadratic: x³ + 7x² - x + 16 = x(x² - 2x + 5) + 9x² - 6x + 16 = 9x² - 6x + 16. Extract again: 9(x² - 2x + 5) + 12x - 29 = 12x - 29. Now substitute x = 1 + 2i: 12(1 + 2i) - 29 = 12 + 24i - 29 = -17 + 24i.

Step 4: Final Answer:
The value is -17 + 24i, option (B).