Algebra Know-It-ALL

413

CHAPTER

25 Cubic Equations in Real Numbers

Let’s move into single-variable cubic equations, also called cubics or third-degree equations. This

type of equation always has a term in which the variable is cubed. There may also be a term

with the variable squared, a term with the variable itself (to the first power), and a stand-alone

constant. We’ll be concerned only with the real-number roots of single-variable cubics having

real coefficients and a real constant.

Cube of Binomial

Some cubics can be expressed as the cube of a binomial with a real coefficient and a real con-

stant. Cubics in this form are easy to solve. They have one real root, which can be derived from

the binomial by setting it equal to zero.

Binomial-cubed form

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

number constant. Consider the expression

(ax+b)^3

If we set this equal to 0, we get

(ax+b)^3 = 0

This is a cubic equation in binomial-cubed form. Here are three examples:

x^3 = 0

(x+ 3)^3 = 0

(2x− 3)^3 = 0

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