Locating a Zero

Boundedness Theorem

Let f(x) be a polynomial function of degree $n\ge1\:$with real coefficients and with a positive leading coefficient. Suppose f(x) is divided synthetically by $x-c$.

1. If $c>0\:$and all numbers in the bottom row of the synthetic division are nonnegative, then f(x) has no zero greater than c.
2. If $c<0\:$and the numbers in the bottom row of the synthetic division alternate in sign (with 0 considered positive or negative, as needed), then f(x) has no zero less than c.

Example

Show that the real zeros of $f\left(x\right)=2x^4-5x^3+3x+1\:$satisfy these conditions:

1. No real zero is greater than 3.
2. No real zero is less than -1.

Solution

1. Since f(x) has real coefficients and the leading coefficient, 2, is positive, use the boundedness theorem. Divide f(x) synthetically by x − 3.
   
   Since 3 > 0 and all numbers in the last row of the synthetic division are nonnegative, f(x) has no real zero greater than 3.
2. Divide by x+1 using synthetic division.
   
   

   Here −1 < 0 and the numbers in the last row alternate in sign, so f(x) has no real zero less than −1.