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 complex roots are involved, try simplifying the expression by using conjugate symmetry or properties of polynomials with real coefficients.

Answer 1

The Correct Option is C

Given: \[ x = -5 + 2\sqrt{-4} = -5 + 4i \] Let’s compute powers of \( x \) up to \( x^4 \) using binomial expansions or software. Alternatively, we recognize a trick: Let’s denote: \[ x = -5 + 4i,\quad \bar{x} = -5 - 4i \] Then, if \( f(x) = x^4 + 9x^3 + 35x^2 - x + 4 \), we can evaluate using the identity \( f(x) + f(\bar{x}) = 2 \cdot \text{Re}(f(x)) \) So, since \( x \) is complex and \( f(x) + f(\bar{x}) \) is real, we can compute just the real part and double it. Using algebraic expansion or substitution (or known evaluation), one finds: \[ f(x) = -160 \quad \Rightarrow \text{Exact value} \]