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

 

Solution:  Answer: (D)

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

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

Therefore, only real root of the equation is \( x=-1\)

 

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