Algebra Know-It-ALL

436 Polynomial Equations in Real Numbers

What are the real roots?

As we search for real roots to a higher-degree equation in binomial factor form, we know we’ve

“made a hit” when any of the factors becomes 0. That means we can set the binomial factors

equal to 0, one by one, and then solve each of them to get all the real roots.

Consider the first of the six binomial factor equations listed above. Let’s find all the real

roots of

(x+ 1)(x− 2)(x+ 3)(x− 4) = 0

We take each binomial individually, set it equal to 0, and then solve the resulting first-degree

equations:

x+ 1 = 0

x− 2 = 0

x+ 3 = 0

x− 4 = 0

Any value of x that satisfies one of these is a root of the higher-degree equation. There are four

such values:

x=−1 or x= 2 or x=−3 or x= 4

so the real solution set of the higher-degree equation is X= {−1, 2, −3, 4}.

Now consider another higher-degree equation. This one is a little tricky, because three of

the four binomials have exponents attached:

(x+ 1)(x− 2)^2 (x+ 3)^3 (x− 4)^4 = 0

If we remove the exponents from the binomials and set each equal to 0, we get the same

four first-degree equations as before. That means the real roots of the higher-degree equa-

tion are the same, too. But three of the four roots have multiplicity greater than 1. The

root x= 2 has multiplicity 2, the root x=−3 has multiplicity 3, and the root x= 4 has

multiplicity 4.

Are you confused?

Does the concept of multiplicity still seem esoteric? In the example we just finished, it can help if we write

out every binomial factor individually so none is raised to a power (other than the first power). Grouped

according to their constants, those factors are

(x+ 1)

(x− 2)(x− 2)

(x+ 3)(x+ 3)(x+ 3)

(x− 4)(x− 4)(x− 4)(x− 4)