Rational Root Theorem
Example 1 Identify Possible Zeros
List all of the possible rational zeros of each function.
a. p(x) = x4 + 4x3 – x2 + 3x – 72
Since the coefficient of x4 is 1, the possible rational zeros must be a factor of the constant term 72. So,
the possible rational zeros are the integers ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, and ±72.
b. h(x) = 3x5 – 2x3 + x + 2
If
p
q
is a rational root, then p is a factor of 2 and q is a factor of 3. The possible values of p are ±1 and
±2. The possible values of q are ±1 and ±3. So all of the possible rational zeros are as follows.
p
q
= ±1, ±2, ±
1
3
, and ±
2
3
.
Example 2 Find Rational Zeros
Find all of the rational zeros for h(x) = x3 – 2x2 – 29x + 30.
The leading coefficient is 1, so the possible integer zeros are factors of 30, ±1, ±2, ±3, ±5, ±6, ±10, ±15,
and ±30. From Descartes’ Rule of Signs, you can tell that there are 2 or 0 positive real zeros. Make a table
and test possible real zeros.
x
1
–2
–29
30
1
1
–1
–30
0
Since (x – 1) is found to be a factor, try to factor the depressed polynomial so that other zeros do not have
to be tested.
x3 – 2x2 – 29x + 30 = (x – 1)(x2 – x – 30)
= (x – 1)(x + 5)(x – 6)
Now, use the Zero Product Property to find all of the zeros.
(x – 1)(x + 5)(x – 6) = 0
x – 1 = 0 or x + 5 = 0 or x – 6 = 0
x = 1 x = –5 x = 6
The rational zeros of this function are –5, 1, and 6.
