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Rational Zeros Theorem:
If the polynomial
has integer coefficients, then every rational zero of P(x) is of the form
where p is a factor of the constant coefficient a0 and q is a factor of the leading coefficient an.
,...)( 011
1 axaxaxaxP nn
nn
,q
p
Example 1:
List all possible rational zeros of P(x) = 2x4 − x2 − 7.
q
p ,1 7,1 2
,1possible rational zeros:
2 of divisors
7 of divisors
,2
1 ,7
2
7
Example 2:
Find the real zeros of f(x) = 2x3 − 5x2 − 4x + 3. Write f(x) in factored form and sketch its graph.
q
p ,1,1 2
,1possible rational zeros:
2 of divisors
3 of divisors
,2
1 ,3
2
3
3
21 5 4
2
2
3
3
7
3
7
4
3 4x - 5x - 2x f(x) 23 ,1possible rational zeros: ,2
1 ,3
2
3
21 5 4
2
2
7
7
3
3
3
0
So -1 is a root which means that
factora is )1(x
)(xQ 13723 452 223 xxxx xx
Consequently, we need only factor
372 2 xx 312 xx
13123 452 23 xxxx xx
Example 3:
Find the real solutions of the equation
06762 234 xxxx
q
p ,1 ,3
1,2
,1possible rational zeros:
1 of divisors
6 of divisors
,2 ,3 6
6
,1possible rational zeros: ,2 ,3 606762 234 xxxx
11 2 6
1
1
3
3
3
7
3
10
6
10
4
11 2 6
1
1
1
1
7
7
7
0
6
0
6
12 2 6
1
2
4
8
2
7
43
6
6
0
possible rational zeros: ,2 ,3 606762 234 xxxx
6762 234 xxxx 324 23 xxx )2(x
2x 324 23 xxx
possible rational zeros: 3324 23 xxx
,1
possible rational zeros: 3324 23 xxx
13 4 2
1
3
7
21
23
3
69
6613 4 2
1
3
1
3
1
3
3
0
6762 234 xxxx 2x 324 23 xxx 3x 2x 12 xx