Polynomials: Bounds on Zeros

Where would you expect to find the roots of a polynomial?.

A Polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

A polynomial has coefficients:

polynomial coefficients
(The terms are in order from highest to lowest exponent)

A polynomial also has roots:

A "root" (or "zero") is where the polynomial is equal to zero.

Example: 3x - 6 equals zero when x=2, because 3(2)-6 = 6-6 = 0

Where are the Roots (Zeros)?

It can sometimes be hard to find where the roots are!

... where should we search ... how far left or right should we go?

Here I will show you a clever way to know where to search for all Real roots.

And it just uses simple arithmetic!


First: prepare your data!

  • The leading coefficient must be 1. If it is not, then divide every term of the polynomial by the leading coefficient.
  • Write down all the coefficients.
  • Then throw away the leading coefficient!
  • Remove minus signs.
  • This is your list of values for the next step

Next: Calculate two different "bounds" using those values

  • Bound 1: Find The largest value and add 1.
  • Bound 2: Add up all the values. If that is greater than 1 then it is your answer. Otherwise the answer is 1.

The smallest "bounds" value is your answer ...

... all roots will be within plus or minus of that!



Example: x3 + 2x2 - 5x + 1

The leading coefficient is 1, so we can continue.

The coefficients are: 1, 2, -5, 1

Drop the leading coefficient, and remove any minus signs: 2, 5, 1

  • Bound 1: the largest value is 5. Plus 1 = 6
  • Bound 2: adding all values is: 2+5+1 = 8

The smallest bound is 6

All Real roots will be between -6 and +6

So we can graph between -6 and 6 and find any Real roots. It is best to plot a little wider so we could see if a curve has roots right at -6 or 6:

polynomial bounds

Now you can just zoom in to the graph to get more accurate values for the roots


Example: 10x5 + 2x3 - x2 - 3

the leading coefficient is 10, so we must divide all terms by 10:

x5 + 0.2x3 - 0.1x2 - 0.3

The coefficients are: 1, 0.2, -0.1, -0.3
Drop the leading coefficient, and remove any minus signs: 0.2, 0.1, 0.3

  • Bound 1: the largest value is 0.3. Plus 1 = 1.3
  • Bound 2: adding all values is: 0.2+0.1+0.3 = 0.6, which is less than 1, so the answer is 1

The smallest is 1.

All Real roots will be between -1 and +1

I will leave the graphing to you.


"Bound 1" and "Bound 2" are not the only ways to find the bounds of the roots, but they are easy to use!

Also Note: Graphing polynomials can only find Real roots, but there can also be Complex roots.