This document discusses finding the rational zeros of polynomials using the Rational Zeros Theorem. It provides examples of finding all rational zeros of polynomials by considering possible values of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. It also discusses using synthetic division and the Quadratic Formula to find the exact zeros of polynomials when not all zeros are rational.

1. 3.3 Real Zeros of
Polynomials
Philippians 4:6-7 do not be anxious about
anything, but in everything by prayer and
supplication with thanksgiving let your requests
be made known to God. And the peace of God,
which surpasses all understanding, will guard
your hearts and your minds in Christ Jesus.

2. Rational Zeros Theorem

3. Rational Zeros Theorem
If P(x) = an x + an−1 x
n n−1
+ an−2 x n−2
+ ... + a1 x + a0
has integral coefficients, then every rational zero
p
of P(x) is of the form where
q
p is a factor of the constant term, and
q is a factor of the leading coefficient.

4. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2

5. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2
p 1 5 7 35
=± , , ,
q 1 1 1 1

6. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2
p 1 5 7 35
=± , , ,
q 1 1 1 1
set the window on your grapher to [-35,35]

7. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2
p 1 5 7 35
=± , , ,
q 1 1 1 1
set the window on your grapher to [-35,35]
graph and test

8. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2
p 1 5 7 35
=± , , ,
q 1 1 1 1
set the window on your grapher to [-35,35]
graph and test
(we are applying the Remainder Theorem here)

9. Find all rational zeros of P(x) = x − 11x + 23x + 35
3 2
p 1 5 7 35
=± , , ,
q 1 1 1 1
set the window on your grapher to [-35,35]
graph and test
(we are applying the Remainder Theorem here)
x = − 1, 5, 7

10. Factor 3x − 4x − 13x − 6
3 2

11. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.

12. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , ,
q 1 1 1 1 3 3 3 3

13. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3

14. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
3

15. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
3x = −2
3x + 2 = 0

16. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
x = −1
3x = −2
3x + 2 = 0 x +1 = 0

17. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
x = −1 x=3
3x = −2
3x + 2 = 0 x +1 = 0 x−3= 0

18. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
x = −1 x=3
3x = −2
3x + 2 = 0 x +1 = 0 x−3= 0
(3x + 2)(x + 1)(x − 3)

19. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
x = −1 x=3
3x = −2
3x + 2 = 0 x +1 = 0 x−3= 0
⎛ 2 ⎞
(3x + 2)(x + 1)(x − 3) Do not use ⎜ x + ⎟
⎝ 3 ⎠

20. Factor 3x − 4x − 13x − 6
3 2
This means we are looking for the zeros.
p 1 2 3 6 1 2 3 6
=± , , , , , , , window: [-6,6]
q 1 1 1 1 3 3 3 3
2
Zeros are: −1, − , 3
2 3
x=−
3
x = −1 x=3
3x = −2
3x + 2 = 0 x +1 = 0 x−3= 0
⎛ 2 ⎞
(3x + 2)(x + 1)(x − 3) Do not use ⎜ x + ⎟
⎝ 3 ⎠

21. 3
Find the exact zeros of f (x) = x − 6x + 4

22. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q

23. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q
graph and test

24. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q
graph and test
x=2

25. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q
graph and test
x=2 but then the other 2 roots
must be irrational

26. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q
graph and test
x=2 but then the other 2 roots
must be irrational
“exact zeros” ... no calculator!

27. 3
Find the exact zeros of f (x) = x − 6x + 4
p
= ± 1, 2, 4 standard window
q
graph and test
x=2 but then the other 2 roots
must be irrational
“exact zeros” ... no calculator!
use synthetic division until it’s a quadratic
then use the Quadratic Formula

28. 3
Find the exact zeros of f (x) = x − 6x + 4

29. 3
Find the exact zeros of f (x) = x − 6x + 4
2 1 0 -6 4
2 4 -4
1 2 -2 0

30. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
2 4 -4
1 2 -2 0

31. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
−2 ± 4 − (4)(−2) 2 4 -4
x=
2 1 2 -2 0

32. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
−2 ± 4 − (4)(−2) 2 4 -4
x=
2 1 2 -2 0
−2 ± 12
x=
2

33. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
−2 ± 4 − (4)(−2) 2 4 -4
x=
2 1 2 -2 0
−2 ± 12
x=
2
−2 ± 2 3
x=
2

34. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
−2 ± 4 − (4)(−2) 2 4 -4
x=
2 1 2 -2 0
−2 ± 12
x=
2
−2 ± 2 3
x=
2
x = −1 ± 3

35. 3
Find the exact zeros of f (x) = x − 6x + 4
2
x + 2x − 2 2 1 0 -6 4
−2 ± 4 − (4)(−2) 2 4 -4
x=
2 1 2 -2 0
−2 ± 12
x=
2
x = 2, − 1 ± 3
−2 ± 2 3
x=
2
x = −1 ± 3

36. Find the exact zeros of P(x) = x + 4x + 3x − 2
3 2

37. Find the exact zeros of P(x) = x + 4x + 3x − 2
3 2
x = −2, − 1 ± 2

38. 4 2
Find all real zeros of f (x) = 10x − x + 4x − 6

39. 4 2
Find all real zeros of f (x) = 10x − x + 4x − 6
Doesn’t say exact ... approximations OK!

40. 4 2
Find all real zeros of f (x) = 10x − x + 4x − 6
Doesn’t say exact ... approximations OK!
p
→ [ −6,6 ] standard window
q

41. 4 2
Find all real zeros of f (x) = 10x − x + 4x − 6
Doesn’t say exact ... approximations OK!
p
→ [ −6,6 ] standard window
q
graphing suggests 2 zeros ... they are:

42. 4 2
Find all real zeros of f (x) = 10x − x + 4x − 6
Doesn’t say exact ... approximations OK!
p
→ [ −6,6 ] standard window
q
graphing suggests 2 zeros ... they are:
x ≈ −1.03, .77
and the other two are imaginary

43. HW #3
“Never doubt that a small group of thoughtful
committed people can change the world; indeed
it is the only thing that ever has.”
Margaret Mead