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 the complex value definition $x = 1 + 2i$, we are tasked with evaluating the numerical value of the higher-degree polynomial expression $x^3 + 7x^2 - x + 16$.

Step 2: Key Formula or Approach:
Rather than calculating complex powers directly, we can isolate the imaginary unit to form a real quadratic equation equal to zero, and use this relation to reduce the degree of the polynomial expression. Given: $x = 1 + 2i \implies x - 1 = 2i$ Square both sides of the equation: $$(x - 1)^2 = (2i)^2$$ $$x^2 - 2x + 1 = -4 \implies x^2 - 2x + 5 = 0$$

Step 3: Detailed Explanation:
Since $x^2 - 2x + 5 = 0$, any term containing this quadratic factor will evaluate to zero. Let's rewrite our target expression by grouping terms to pull out this quadratic base: $$x^3 + 7x^2 - x + 16 = x(x^2 - 2x + 5) + 9x^2 - 6x + 16$$ Substitute $x^2 - 2x + 5 = 0$: $$= x(0) + 9x^2 - 6x + 16 = 9x^2 - 6x + 16$$ Now, break down $9x^2 - 6x + 16$ further using another multiple of our zero-valued quadratic base: $$9x^2 - 6x + 16 = 9(x^2 - 2x + 5) + 12x - 29$$ Substitute $x^2 - 2x + 5 = 0$ once more: $$= 9(0) + 12x - 29 = 12x - 29$$ The original expression simplifies down to the linear expression $12x - 29$. Now, substitute the original value $x = 1 + 2i$ back into this simplified form: $$\text{Value} = 12(1 + 2i) - 29$$ $$\text{Value} = 12 + 24i - 29 = -17 + 24i$$

Step 4: Final Answer:
The total value of the polynomial expression is $-17 + 24i$, which corresponds directly to option (B).