Finding Rational Roots
Steps of Rational Zeros
1. Identify the constant term and coefficient
of the polynomial
2. List all factors of each number and don’t
forget the plus-or-minus symbol for each
term
3. Apply the equation and DIVIDE
p
Factors of Constant Term

q Factors of Leading Coefficient
4. List ALL of the possible rational zeros
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5.6 – Find Rational Zeros
2
Example 1
List all possible rational zeros for x8  18x 4  9 x 2  2  0
Constant: 2
Leading Coefficient: 1
p +1,  1, 2,  2

q
+1,  1
p
 1, 2
q
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2
1
1
2
 1
 1   2   2
1
1
1
1
1
1
2
2
 1,  1,  2,  2
1
1
1
1
5.6 – Find Rational Zeros
3
Example 2
List all possible rational zeros for
f ( x)  2 x 4  x 3  17 x 2  4 x  6
Constant: 6
Leading Coefficient: 2
p +1,  1, 2,  2, +3,  3, +6,  6

q
+1,  1, +2,  2
1
3
p
 1,  2,  3,  6,  , 
q
2
2
p
1 3
 1, 2, 3, 6,  , 
q
2 2
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5.6 – Find Rational Zeros
4
Steps in finding rational zeros
A. Determine what the leading coefficient and
constant term are.
B. Put it in the rational zero test (p/q).
C. Use synthetic division to find which factor
gives the remainder of zero.
If the polynomial cannot be factored OR it is
NOT a QUADRATIC, repeat synthetic division.
D. Factor or use the Quadratic Formula
E. Equal it all to zero and put it in x= form.
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5.6 – Find Rational Zeros
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Example 3
3
2
f
(
x
)

x

7
x
 11x  5
Find the roots for
p +1,  1, 5,  5

q
+1,  1
p
 1,  5
q
7
1 1
11
1 6
1 6 5
5
5
0
X-1 IS A factor
x  6x  5  0
 x  1 x  5  0
2
 x  1  x  1 x  5  0
x 1 x 1 x  5
x  1 with a multiplicity of 2 and x  5
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5.6 – Find Rational Zeros
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Example 4 Find the roots for the function…
f ( x)  3x 3  7 x 2  22 x  8
p +1,  1, 2,  2, +4,  4, +8,  8

q
+1,  1, +3  3
p
1 2 4 8
 1,  2,  4,  8,  ,  ,  , 
q
3 3 3 3
1 3
7
 22
10
3
3 10 12
X-1 is NOT a factor
8
12
20
1 3
7
3
3
4
 22
4
26
8
26
18
X+1 is NOT factor
2 3
3
7
6
13
 22
26
4
8
8
0
X-2 IS A factor
3x 2  13x  4  0
1 12
3x 2  12 x  1x  4  0 2 6
2
3
x
  12x   1x  4  0 3 4
3x  x  4  1 x  4   0
 x  4  3x  1  0
7
Example
3
2
f
(
x
)

3
x

7
x
 22 x  8
4 Continued
 x  2  x  4  3x  1  0
x20
x2
x40
x  4
1
x  2, x  4 and x  
3
3x  1  0
3x  1
1
x
3
Example 5 Find the roots for the function…
f ( x)  5 x 3  7 x 2  x  3
p +1,  1, 3,  3

q +1,  1, +5  5
p
1 3
 1,  3,  , 
q
5 5
1 5
7
1
12
5
5 12 11
X-1 is NOT a factor
3
11
14
1
5
5
7
5
2
1
2
3
3
3
0
X+1 IS A factor
1 15
5x 2  2 x  3  0
2
5 x  5 x  3x  3  0 3 5
2
 5x  5x   3x  3  0
5x  x  1 3  x  1  0
 x  1 5x  3  0
 x  1 x  1 5x  3  0
x  1 w / a multiplicity of 2, x 
3
5
Example 6 Find the roots for the function…
f ( x)  x 4  x 3  3 x 2  x  2
p +1,  1, 2,  2

q
+1,  1
p
 1,  2
q
1 1 1  3 1 2
1 2 1 2
1 2 1 2 0
X-1 IS A factor
x  2x  x  2  0
3
2
1
1
2
1
1
1
1
1
2
2
2
0
X+1 IS A factor
x2  x  2  0
 x  1 x  2   0
 x  1 x  1 x  1 x  2   0
x  1 x  1 x  1 x  2
x  1 w / a multiplicity of 2,
x  1, x  2