How many real roots does this polynomial have?

Algebra Level 3

How many real roots does the polynomial x 8 + x 6 + x 4 + x 2 − 1 x^8+x^6+x^4+x^2-1 have?


The answer is 2.

Thiruvikraman Kandhadai by Thiruvikraman Kandhadai , 51, India

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1 solution

Since there are only even powers of x in the given polynomial, excluding the constant term would make it positive for all real values of x. The constant term, -1 simply shifts the graph of ( x 8 + x 6 + x 4 + x 2 (x^8+x^6+x^4+x^2 downwards. Therefore there will be two real roots as shown in the figure given below:

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