Algebra Know-It-ALL

414 Cubic Equations in Real Numbers

In the first case, the stand-alone constant is 0. If we want to be formal, we can write

(x+ 0)^3 = 0

In the second case, the coefficient of x in the binomial is 1 and the stand-alone constant is 3.

In the third case, the coefficient of x in the binomial is 2, and the constant is −3.

Multiplying out

Now look at the last equation shown in the above set of three. We can multiply it out so the

left side becomes a polynomial. First, we can rewrite it as

(2x− 3)(2x− 3)(2x− 3) = 0

Then we can multiply the second two factors using the product of sums rule. (Remember that

subtraction is the same as the addition of a negative.) The result is

(2x− 3)(4x^2 − 12 x+ 9) = 0

If we multiply again using the expanded product of sums rule, consolidate the terms for x^2

andx, and pay close attention to the signs, we get

8 x^3 − 36 x^2 + 54 x− 27 = 0

What’s the real root?

A single-variable equation in binomial-cubed form has one real root, assuming the coefficient

and the constant are both real numbers. That root can be found by taking away the exponent

from the binomial, and then solving the first-degree equation that remains. In general, if we

have the third-degree equation

(ax+b)^3 = 0

then the real root is the solution to

ax+b= 0

That solution is obtained by subtracting b from both sides, and then dividing through by a.

We can get away with division by a, because a≠ 0. Therefore, the single real root to the cubic is

x=−b/a

Are you confused?

Simple cubics can sometimes look complicated, and complicated cubics can sometimes look simple. We

often cannot know by glancing at a third-degree equation whether solving it will be easy, challenging, or

difficult. The following example can illustrate.