Find a bound on the real zeros of the polynomial function.
-$$f ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 4 x - 1$$
A) $- 15$ and 15
B) $- 6$ and 6
C) $- \frac { 5 } { 3 }$ and $\frac { 5 } { 3 }$
D) $- 10$ and 10

Question

Quizplus · Accepted Answer

The bounds for the real zeros of a polynomial function can be found using the Rational Root Theorem, which suggests that all real zeros are within the range $[-M, M]$, where $M$ is the maximum of $1$ and the ratio of the absolute value of the leading coefficient to the smallest coefficient among the rest of the terms. In this case, the coefficients are $1, 3, 5, -2, 4, -1$, so the maximum absolute value among the coefficients other than the leading one is $5$, and the leading coefficient is $1$. Therefore, the bounds are $[-5, 5]$, but since the question does not provide this option, the closest provided option is $[-6, 6]$, which sufficiently covers the possible range of real zeros.

Find a Bound on the Real Zeros of the Polynomial f(x)=x5+3x4+5x3−2x2+4x−1f ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 4 x - 1f(x)=x5+3x4+5x3−2x2+4x−1

Find a Bound on the Real Zeros of the Polynomial $f ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 4 x - 1$