Polynomials Unit Rational Root Theorem (finding all rational factors of a 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