Rational Zero
Theorem
Algebra II A
Chapter 8
K. Couture
Rational Zero Theorem
• Rational Zero Theorem:
n
n 1
a1 x
..... an 1 x an
Let f(x)= a0 x
represent a polynomial function. If p is a
q
rational number in simplest form and is
a zero of f(x), then p is a factor of an and
q is a factor of a0 .
• Example: List all possible rational zeros for
the function and state whether they are
positive or negative.
f ( x) 3x
1, 5
4
2x
3
5
1, 3
1
5
1,
, 5,
3
3
f ( x) 3x4 2 x3 5
One sign change One positive root
Look at the ‘p’ term or the
constant function and list it’s
possible factors.
Look at the ‘q’ term or the
leading coefficient and list it’s
possible factors.
List the factors in
form.
p
q
To determine the number of
positive and negative roots, use
Descartes Rule.
f ( x) 3( x)4 2( x)3 5
f ( x) 3x4 2x3 5 On sign change = One negative root
4
3
So, if f ( x) 3x 2 x 5 has one positive
root, and one negative root, and all possible
1
5, then use
rational roots are
1,
3
, 5,
3
synthetic division to find a positive or
negative real root.
13
2 0 0
5
1 1 1
3 1 1 1 4
13
3
1 is not a factor…. Try again
3
2 0 0
5
5
3 5
5
5 5
5 0
Most excellent … -1 is a factor
Find all rational zeros for the function f ( x)
Factors for p
1, 28, 2, 14, 4, 7
Factors for q
1
p
q
1, 28, 2, 14, 4, 7
Start Dividing !!
11 4
25
1 5
1 5
20
Determine the number
of positive and
negative roots.
f ( x)
x3 4 x2 25x 28
f ( x)
x3 4x2 25x 28
One positive root / 2 or 0 negative roots
28
20
48
x3 4 x2 25x 28
11 4
1
1 3
25
28
Now use your new
quotient to continue
your division
3 28
28 0
You have two of the three roots, now find the
remaining positive root.
-1, -7, and 4 are the zeros.
7 1 3 28
7 28
1 4 0
Find all of the zeros of each function f ( x)
6 x3 5 x 2 9 x 2
1, 2 List all q factors
List all p factors
List all p factors 1, 2, 1 , 1 , 1 , 2
q
2 3 6 3
1, 2, 3, 6
Determine the number of positive and negative zeros
f ( x) 6 x
3
5x
2
9x 2
f ( x)
1 negative zero
2 or 0 positive zeros
Positive
0
2
6 x3 5 x 2 9 x 2
Negative
1
1
Imaginary
2
0
Now you can start dividing
16 5
6
16 5
9 2
6
11
2
1
2
4
2
6 5 9 2
3
4 6 2
9 2
6 1 8
6
1 -8 10
6 9
3 0
2/3 is a factor – now try to factor the remaining quotient
6 x2 9 x 3 0
Not factorable to help find zeros, so
use the quadratic formula.
3(2 x2 3x 1) 0
6 x2 9 x 3 0
(9)
(9) 2 4(6)( 3)
9
=
2(6)
9 3 17
=
12
=
3
4
81 72
9 153
=
12
12
17 So our zeros are 2/3,
and
=
3
17
4