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

Question

Quizplus · Accepted Answer

The real zeros of the polynomial function $f(x) = x^4 + 2x^2 - 3$ are bounded by the values that make the polynomial equal to zero. To find a bound, we can use the fact that for a polynomial of the form $x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, a bound on the real zeros can be given by $\pm(1 + \max\{|a_{n-1}|, |a_{n-2}|, \ldots, |a_0|\})$. In this case, the largest coefficient in absolute value (other than the leading coefficient) is 3. Therefore, a bound on the real zeros is $\pm(1 + 3) = \pm 4$, which corresponds to choice C.

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

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