The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.

1. RATIONAL ROOT
THEOREM
Unit 6: Polynomials

2. Finding All Factors

3. Recap
 We can use the Remainder & Factor
Theorems to determine if a given linear
binomial (𝑥 − 𝑐) is a factor of a polynomial
𝑓(𝑥).
 Remember: (𝑥 − 𝑐) is a factor of 𝑓(𝑥) if and only if
𝑓(𝑐) = 0. In other words, the remainder after
synthetic division must be zero in order for the
linear binomial to be a factor of the polynomial.
 Example: Prove (𝑥 + 2) is a factor of
𝑓 𝑥 = 3𝑥3
− 4𝑥2
− 28𝑥 − 16
 𝑓 −2 = 3(−2)3
−4 −2 2
− 28(−2) − 16
 𝑓 −2 = 3(−8) − 4(4) − 28(−2) − 16
 𝑓 −2 = −24 − 16 + 56 − 16 = 0

4. How do we find the other
factors?
 The quotient we get after synthetic division is
called the depressed polynomial
 We can FACTOR this depressed polynomial!!!
 Factor this polynomial using a previously learned
method!
 GCF
 Simple Case (multiply to “c” & add to “b”)
 Slide & Divide

5. Example
 We have already proved (𝑥 + 2) is a factor of
(3𝑥3
− 4𝑥2
− 28𝑥 − 16).
 We can find the depressed polynomial from
synthetic division.
 The depressed polynomial (quotient) is (3𝑥2
−
10𝑥 − 8)

6. Example (Cont.)
 Now, lets factor our depressed polynomial
using Slide & Divide…
 3𝑥2
− 10𝑥 − 8
 𝑥2
− 10𝑥 − 24
 𝑥 − 12 𝑥 + 2
 𝑥 −
12
3
𝑥 +
2
3
 (𝑥 − 4)(3𝑥 + 2)

7. Example (Cont.)
 Therefore, all of the factors of our original
trinomial 𝑓 𝑥 = 3𝑥3
− 4𝑥2
− 28𝑥 − 16 are:
 (𝑥 + 2)(𝑥 − 4)(3𝑥 + 2)

8. Try #’s 1-4 from your worksheet on your
own!
PRACTICE

9. What happens when you must factor a
polynomial of degree ≥ 3 and you do not
know any factors?!
Rational Root Theorem

10. Rational Root Theorem
 If 𝑓 𝑥 = 𝑎 𝑛 𝑥 𝑛
+ ⋯ + 𝑎1 𝑥1
+ 𝑎0 has integer
coefficients, then every rational zero of
𝑓(𝑥) has the following form:

𝑝
𝑞
=
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚 𝑎0
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑎 𝑛

11. Example
 Find all possible rational roots of 𝑓(𝑥) using
the Rational Root Theorem
 𝑓(𝑥) = 4𝑥4
− 𝑥3
− 3𝑥2
+ 9𝑥 − 10
 Factors of the constant term: ±1, ±2, ±5, ±10
 Factors of the leading coefficient: ±1, ±2, ±4
 Possible rational zeros:
±
1
1
, ±
2
1
, ±
5
1
, ±
10
1
, ±
1
2
, ±
2
2
, ±
5
2
, ±
10
2
, ±
1
4
, ±
2
4
, ±
5
4
, ±
10
4
 Simplified list: ±1, ±2, ±5, ±10, ±
1
2
, ±
5
2
, ±
1
4
, ±
5
4

12. Try #’s 5-8 from your worksheet on your
own!
PRACTICE

13. Example 1
 Find all the rational roots of the given function
𝑓 𝑥 = 𝑥3
− 4𝑥2
− 11𝑥 + 30
 Possible Zeros: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30
 Check using Remainder & Factor Theorems:
 Since 2 gives a 0 remainder, that means (𝑥 − 2)
is a factor.

14. Example 1 (Cont.)
 Now, use synthetic division to find the
depressed polynomial
 Factor (𝑥2
− 2𝑥 − 15) to find the remaining
factors
 𝑥2 − 2𝑥 − 15 = (𝑥 − 5)(𝑥 + 3)
 Therefore, all the factors are:
 (𝑥 − 2)(𝑥 − 5)(𝑥 + 3)

15. Example 2
 Find all the rational roots of the given function
𝑓 𝑥 = 2𝑥4
− 5𝑥3
− 28𝑥2
+ 15𝑥
 Notice that this polynomial has a GCF of x!
 Factor out the GCF: 𝑓 𝑥 = 𝑥(2𝑥3 − 5𝑥2 − 28𝑥 +
15)
 Possible Zeros: ±1, ±3, ±5, ±15, ±
1
2
, ±
3
2
, ±
5
2
, ±
15
2
 Check using Remainder & Factor Theorems:

16. Example 2 (Cont.)
 Since −3 gives a 0 remainder, that means (𝑥 +
3) is a factor.
 Now, use synthetic division to find the depressed
polynomial
 Factor (2𝑥2
− 11𝑥 + 5) to find the remaining
factors
 2𝑥2 − 11𝑥 + 5 = (𝑥 − 5)(2𝑥 − 1)
 Therefore, all the factors are:
 𝑥(𝑥 + 3)(𝑥 − 5)(2𝑥 − 1)

17. Try #’s 9-10 from your worksheet on your
own!
PRACTICE