If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form ±
p/q, where
p is a factor of the constant term and
q is a factor of the leading coefficient.
Example 1: Find all the rational zeros of
According to the rational zero theorem, any rational zero must have a factor of 3 in the numerator and a factor of 2 in the denominator.
The possibilities of
p/q, in simplest form, are
These values can be tested by using direct substitution or by using synthetic division and finding the remainder. Synthetic division is the better method because if a zero is found, the polynomial can be written in factored form and, if possible, can be factored further, using more traditional methods.
Example 2: Find rational zeros of 2
x3 + 3
x2 − 8
x + 3 by using synthetic division.
The zeros of
f (
x)=2
x3 + 3
x2 − 8
x + 3 are 1, ½, and −3. This means
The zeros could have been found without doing so much synthetic division. From the first line of the chart, 1 is seen to be a zero. This allows
f(
x) to be written in factored form using the synthetic division result.
But 2
x2 + 5
x − 3 can be further factored into (2
x − 1)(
x + 3) using the more traditional methods of factoring.
From this completely factored form, the zeros are quickly recognized. Zeros will occur when
Linear Sentences in One Variable
Polynomial Functions
