Question

Find the number of rational roots of the equation $2x^{99}+3x^{98}+2x^{97}+...+2x+3=0$.

• A. $2$
• B. $1$
• C. $3$
• D. $0$

Answer 1

Answer: (B)

$1$

$P\left(x\right)=2x^{99}+3x^{98}+2x^{97}+\cdots +2x+3=0$

$\Rightarrow P\left(x\right)=(2x+3)(x^{98}+x^{96}+x^{94}+\cdots +1)$

$\Rightarrow P\left(x\right)=(2x+3)(x^2+1)(x^{96}+x^{92}+x^{88}+\cdots +1)$

Also,

$(x^{96}+x^{92}+x^{88}+\cdots +1)>0\forall x\in R$

$\therefore 2x+3=0$ or $x^2+1=0$

$\Rightarrow x=-\frac{3}{2}$ or $x=\pm i$

Only one real solution exists.