If $x = -5 + 2\sqrt{-4}$, find the value of $x^4 + 9x^3 + 35x^2 - x + 4$.

Question

Accepted Answer

Here $x = -5 + 4i$. Then $x + 5 = 4i$. Squaring both sides: $(x+5)^2 = 16i^2 \implies x^2 + 10x + 25 = -16 \implies x^2 + 10x + 41 = 0$. Dividing the polynomial $x^4 + 9x^3 + 35x^2 - x + 4$ by $x^2 + 10x + 41$ gives a quotient of $x^2 - x + 4$ and a remainder of 0. Thus, the value is 0.

Complex Numbers and Quadratic Equations

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