Algebra Know-It-ALL

422 Cubic Equations in Real Numbers

Polynomial Equation of Third Degree

Any single-variable cubic equation can be written so it appears in polynomial standard form.

Often, this is the form you’ll first see.

Polynomial standard form

When a cubic equation is in polynomial standard form, the left side of the equals sign con-

tains a third-degree polynomial, and the right side contains 0 alone. Here’s the general form

in the realm of the real numbers:

ax^3 +bx^2 +cx+d= 0

where a, b, c, and d are real numbers, a≠ 0, and x is the variable. All of the following equations

are cubics in this form:

x^3 + 3 x^2 + 2 x+ 5 = 0

− 2 x^3 + 3 x^2 − 4 x= 0

5 x^3 − 7 x^2 − 5 = 0

− 7 x^3 − 4 x^2 = 0

4 x^3 + 3 x= 0

− 7 x^3 − 6 = 0

If you set a= 0 in the polynomial standard form of a single-variable cubic equation, you end

up with the polynomial standard form for a single-variable quadratic, or even a first-degree

equation (depending on the other coefficients). However, the coefficients of x 2 or x, as well as

the stand-alone constant, can equal 0 in a cubic polynomial.

What can we do?

When we see a cubic equation in polynomial standard form, the roots may be apparent right

away, but often they are not. If we come across the equation

x^3 − 8 = 0

we can see, perhaps without having to do any manipulations, that there’s one real root, x= 2.

But if we encounter

x^3 + 3 x^2 + 2 x+ 5 = 0

the situation is more challenging. When we see an equation like this, we can try to factor it

into a product of binomials, or into binomial-trinomial form. If we can manage to do that,

then we can find the roots as described in the previous sections. But factoring a cubic polyno-

mial is not always easy. If the coefficients and the constant of the polynomial equation are all

real numbers, a binomial-trinomial expression for the equation must exist, but the coefficients

and constants might not be integers. They might even turn out to be irrational numbers.