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Roots of a Quintic Polynomial | TOMATO Objective 257

Try this beautiful problem from TOMATO Objective no. 257 based on Roots of a Quintic Polynomial.

Problem: Roots of a Quintic Polynomial

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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