Applications of Derivatives
Roots of Polynomial — Product of Complex Roots
nta_pyq_2023_jan
Grade 12
Question:
Let $\alpha_1,\alpha_2,\ldots,\alpha_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $|\alpha_1|\geq|\alpha_2|\geq\cdots\geq|\alpha_7|$. Then $\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$ is equal to ___.
Step-by-Step Solution
Key Concept: Factor: $x(x^6+3x^4-13x^2-15)=0$. Substituting $t=x^2$: $t^3+3t^2-13t-15=0\Rightarrow(t-3)(t^2+6t+5)=0\Rightarrow t=3,-1,-5$. Roots: $x=0,\pm\sqrt{3},\pm i,\pm\sqrt{5}i$.
$\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6=5-(-3)+1=9$.
Correct Answer: 9