Question

If \[ x=-5+2\sqrt{-4}, \] then the value of \[ x^4+9x^3+35x^2-x+4 \] is:

• A. \(80\)
• B. \(160\)
• C. \(-160\)
• D. \(-80\)

Hint:

When a complex number satisfies a quadratic equation, reduce higher powers using the quadratic relation instead of expanding directly.

Answer 1

The Correct Option is C

Step 1: Simplify $x$.
$x = -5 + 2\sqrt{-4} = -5 + 2(2i) = -5 + 4i$. So $x + 5 = 4i$.

Step 2: Find the minimal polynomial of $x$.
Square both sides: $(x+5)^2 = (4i)^2 = -16$, so $x^2 + 10x + 25 = -16$, giving $x^2 + 10x + 41 = 0$. This means $x^2 = -10x - 41$ for our specific $x$.

Step 3: Express $x^3$ and $x^4$ using the minimal polynomial.
$x^3 = x \cdot x^2 = x(-10x-41) = -10x^2 - 41x = -10(-10x-41)-41x = 100x+410-41x = 59x+410$.
$x^4 = x \cdot x^3 = x(59x+410) = 59x^2+410x = 59(-10x-41)+410x = -590x-2419+410x = -180x-2419$.

Step 4: Substitute into the expression.
\[ x^4+9x^3+35x^2-x+4 \] \[ = (-180x-2419) + 9(59x+410) + 35(-10x-41) - x + 4 \] \[ = -180x-2419+531x+3690-350x-1435-x+4 \]

Step 5: Collect like terms.
$x$ terms: $(-180+531-350-1)x = 0x = 0$. Constant terms: $-2419+3690-1435+4 = -160$.

Step 6: State the answer.
\[ \boxed{-160} \]