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# Roots of a Quintic Polynomial (TOMATO Objective 257)

Problem: The number of real roots of $$x^5+2x^3+x^2+2=0$$ is

(A) 0

(B) 3

(C) 5

(D) 1

$$x^5+2x^3+x^2+2=0$$ $$\implies x^3(x^2+2)+(x^2+2)=0$$ $$\implies (x^3+1)(x^2+2)=0$$ $$\implies (x+1)\bold{\underline{(x^2-x+1)(x^2+2)}}=0$$

The expression in underline doesn’t have any real roots.

Therefore, only real root of the equation is $$x=-1$$

January 2, 2017