If we see a cubic in polynomial standard form and we find ourselves staring at it, para-
lyzed with uncertainty, there’s some good news, some fair news, and some bad news. First, the
good news: We can try to factor a binomial out of a polynomial using a process called synthetic
division. That often works, but we’ll probably have to go through the process several times
before we find the right binomial. Now for the fair news: A general formula for solving a cubic
equation exists. Finally, the bad news: That formula is so complicated that most mathemati-
cians would rather try every other option first.
Before we see how synthetic division works, we should formalize an important principle.
The binomial factor rule
A little while ago, I made a strong claim: Any cubic equation with real coefficients and a real
constant has at least one real root. Suppose we call that real number k. If we plug in k for the
variable in the equation—no matter what form that equation happens to be in—and work
out the arithmetic, the result will be 0.
Now imagine that we are faced with a cubic equation in polynomial standard form, and
we can’t figure out how it can be factored. It seems reasonable to suppose that if we can find
the real root k, which must exist, then the binomial (x−k) can be factored out of the cubic.
That’s because if we set x equal to k, then the value of (x−k) becomes 0; and with 0 as one of
the factors, the whole expression attains the value 0. We can package all this reasoning up into
a formal statement called the binomial factor rule or the factor theorem:
- A real number k is a root of a cubic equation in the variable x if and only if (x−k) is a
factor of the cubic polynomial.
Remember what “if and only if ” means in logical terms: the “if-then” reasoning works
both ways. We can therefore rewrite the above rule as two separate statements:
- If a real number k is a root of a cubic equation in the variable x, then (x−k) is a factor
of the cubic polynomial. - If k is a real number and (x−k) is a factor of a cubic polynomial in the variable x, then
k is a real root of the cubic equation.
This rule, which has been proven as a theorem by mathematicians, can be generalized to
polynomial equations in the fourth degree (called quartic equations), the fifth degree (called
quintic equations), and, in fact, any positive-integer degree (called nth-degree equations).
We try, we fail
Now that we’re armed with plenty of theoretical facts, it’s time to take aim at a real polynomial
cubic equation using synthetic division. Let’s try this:
x^3 − 7 x− 6 = 0Remember the general form:
ax^3 +bx^2 +cx+d= 0Polynomial Equation of Third Degree 423