Digging for Real Roots
Now that we’ve learned how to recognize a polynomial equation, it’s time to think about
finding the real roots of such an equation, if any exist. The next few sections offer some ways
to look for the roots of higher-degree equations. But there are no guarantees. Anyone but the
purest mathematician would likely concede that these types of situations lend themselves to
computer programming.
A prefabricated problem
To illustrate how the real roots can be sought when we’re confronted with a polynomial equa-
tion, let’s look for the real roots of
x^4 − 2 x^3 − 13 x^2 + 14 x+ 24 = 0
This equation was built up from factors. Here’s what it looks like in binomial factor form:
(x+ 1)(x− 2)(x+ 3)(x− 4) = 0
We solved this awhile ago. The roots are
x=−1 or x= 2 or x=−3 or x= 4
Now imagine that we’re looking at the polynomial version of this equation, and we’ve never
seen it in any other form. We’ve been told to find the real roots.
How many roots?
When you embark on a quest to find the real roots of a polynomial equation, you might
wonder how long you should keep trying before you give up (or let your computer take over).
In part, it depends how much time and patience you have. Here’s an important principle to
keep in mind:
- A polynomial equation can never have more roots than its degree. That includes not
only the real roots, but all of the complex roots.
If you’re working on a polynomial equation of degree n and you’ve found n roots, you can
terminate your quest. You’ve resolved the mystery. There is nothing more to find.
Bounds for real roots
The real roots of a polynomial equation always lie between two extremes. We can identify an
interval that contains all the elements in the real solution set if we can discover an upper bound
that’s big enough, and if we can discover a lower bound that’s small enough. In this context,
we consider only non-inclusive bounds. That means an upper bound must be greater than the
largest real root of the equation, and a lower bound must be less than the smallest real root of
the equation.
Digging for Real Roots 439
