Algebra Know-It-ALL

432

CHAPTER

26 Polynomial Equations in Real Numbers

In this chapter, we’ll examine some ways to look for real roots of higher-degree equations with

real coefficients and real constants. In this context, the term higher-degree applies to any equa-

tion of degree 4 or more. The tactics described in the last part of this chapter can be helpful in

finding the real roots of stubborn third-degree (cubic) equations as well.

Binomial to the nth Power

Sometimes, a higher-degree equation can be written as a positive-integer power of a binomial

with a real coefficient and a real constant. Such an equation always has exactly one real root,

which can be found by setting the binomial equal to 0 to get a first-degree equation. The root

has multiplicity equal to the value of the power to which the binomial is raised. That’s the

same as the degree of the equation.

The general form

Suppose x is a variable, a is the nonzero real-number coefficient of x, and b is a real-number

constant. Consider the expression

(ax+b)n

where n is a positive integer. If we set this equal to 0, we obtain

(ax+b)n= 0

which is an equation in binomial to the nth form. (We don’t have to include the word “power”

because it’s understood.) Here are some examples of binomial to the nth equations, where n > 3:

(2x− 3)^4 = 0

(6x+ 1)^5 = 0

(23x+ 77)^6 = 0

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