Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding - Page 387: 32

The maximum number of real zeros is $6$ The number of positive real zeros is $0$ The number of negative real zeros is $0$

Work Step by Step

The number of zeros of a polynomial can’t be greater than its degree. $1)$ The number of positive real zeros of $f(x)$ either equals the number of variations in the sign of the nonzero coefficients of $f(x)$ or equals that number minus an even integer $2)$ The number of negative real zeros of $f(x)$ either equals the number of variations in the sign of the nonzero coefficients of $f(-x)$ or equals that number minus an even integer. So the maximum number of real zeros here is $6$ Since $$f\left( x\right) =x^{6}+1$$ has $0$ variations The number of positive real zeros is $0$ Since $$f\left( -x\right) = x^{6}+1$$ Has $0$ variations The number of negative real zeros is $0$.

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