This SIM was crafted to address the needs of learners with difficulties in factoring polynomials.

1. or
Department of Education
Region III
Schools Division of Zambales
District of Masinloc
TALTAL NATIONAL HIGH SCHOOL
Masinloc, Zambales
STRATEGIC INTERVENTION MATERIALS (SIM)
(FACTORS OF POLYNOMIAL)
IN
GRADE 10 MATHEMATICS
Prepared by:
ROCHELLE E. OLIVA
Teacher I

2. LEAST MASTEREDSKILL:
Identifying the factored form
of polynomial.

3. (FACTOR THEOREM)
• The polynomial 𝑃(𝑥) has x-r as a factor if and
only if 𝑃 𝑟 = 0
TWO PARTS OF THE PROOF OF FACTOR THEOREM
• If (𝑥 − 𝑟) is a factor of 𝑃(𝑥), then 𝑃 𝑟 = 0
• If 𝑃 𝑟 = 0, then (𝑥 − 𝑟) is a factor of 𝑃(𝑥).
If 𝑃 𝑟 ≠ 0, then 𝑥 − 𝑟 is a factor of 𝑃(𝑥)
𝒙 − 𝒓 𝒊𝒔 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒇 𝑷 𝒙 𝒊𝒇 𝒂𝒏𝒅 𝒐𝒏𝒍𝒚 𝒊𝒇 𝑷 𝒓 = 𝟎

4. (SOLVING)
EXAMPLE 1. Show that (𝑥 − 1) is a
factor of 3𝑥3
− 8𝑥2
+ 3𝑥 + 2.
Solution: Using the factor theorem, we
have:
𝑟 = 1
𝑃 1 = 3(1)3 − 8(1)2 + 3 1 + 2
𝑃 1 = 3 1 − 8 1 + 3 + 2
𝑃 1 = 3 − 8 + 3 + 2
𝑃 1 = 0
Since P(1)=0, then (x-1) is a factor of
𝟑𝒙 𝟑
− 𝟖𝒙 𝟐
+ 𝟑𝒙 + 𝟐.
EXAMPLE 2. Show that (𝑥 + 2)
is a factor of 5𝑥2 − 2𝑥 + 1.
Solution:
𝑟 = −2
𝑃 −2 = 5(−2)2 − 2 −2 + 1
𝑃 −2 = 5 4 + 4 + 1
𝑃 −2 = 20 + 4 + 1
𝑃 −2 = 25
Since P(-2)=25, (x+2) is a NOT a
factor of 𝟓𝒙 𝟐 − 𝟐𝒙 + 𝟏.

5. (APPLYING THE RATIONAL ROOT THEOREM
AND FACTOR THEOREM IN FINDING THE
FACTORS OF POLYNOMIAL)
EXAMPLE. Write 𝒙 𝟐 − 𝟓𝒙 + 𝟔 = 𝟎 in factored form.
STEPS:
1. Identify the number of roots based on the degree of the
polynomial.
2. Determine the leading coefficient and the constant term.
3. Find the factors of the leading coefficient and as well as the
factors of the constant term.
4. Divide the factors of the constant term by the factors of the
leading coefficient. The answers/ quotients will be used as the
values of r.
5. Identify the factors using the FACTOR THEOREM.

6. (APPLYING THE RATIONAL ROOT THEOREM
AND FACTOR THEOREM IN FINDING THE
FACTORS OF POLYNOMIAL)
EXAMPLE. Write 𝒙 𝟐 − 𝟓𝒙 + 𝟔 = 𝟎 in factored form.
Solution #1: 𝑟 = 1
𝑷 𝟏 = (𝟏) 𝟐
−𝟓 𝟏 + 𝟔
𝑃 1 = 1 − 5 + 6
𝑃 1 = 2
Since P(1) ≠0, 𝑥 − 1 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Solution #2: 𝑟 = −1
𝑷 −𝟏 = (−𝟏) 𝟐
−𝟓 −𝟏 + 𝟔
𝑃 −1 = 1 + 5 + 6
𝑃 −1 = 12
Since P(-1)≠0, 𝑥 + 1 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Solution.
1. The degree of the polynomial is 2, thus the
equation has at most 2 real roots.
2. Leading Coefficient = 1
Constant Term = 6
3. Factors of the leading coefficent are ±1.
Factors of the constant term are ±1, ±2, ±3,
±6.
4. 𝑟 =
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑟 =
±1
±1
; 𝑟 =
±2
±1
; r=
±3
±1
; 𝑟 =
±6
±1
;
Therefore the values of 𝒓 are ±1, ±2, ±3 and
±6
Solution #3: 𝑟 = 2
𝑷 𝟐 = (𝟐) 𝟐
−𝟓 𝟐 + 𝟔
𝑃 2 = 4 − 10 + 6
𝑃 2 = 0
Since P(2)=0, 𝑥 − 2 𝑖𝑠 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.

7. (APPLYING THE RATIONAL ROOT THEOREM
AND FACTOR THEOREM IN FINDING THE
FACTORS OF POLYNOMIAL)
EXAMPLE. Write 𝒙 𝟐
− 𝟓𝒙 + 𝟔 = 𝟎 in factored form.
Solution #5: 𝑟 = −3
𝑷 −𝟑 = (−𝟑) 𝟐
−𝟓 −𝟑 + 𝟔
𝑃 −3 = 9 + 15 + 6
𝑃 −3 = 30
Since P(-3)≠0, 𝑥 + 3 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Solution #6: 𝑟 = 3
𝑷 𝟑 = (𝟑) 𝟐
−𝟓 𝟑 + 𝟔
𝑃 3 = 9 − 15 + 6
𝑃 3 = 0
Since P(3)=0, 𝑥 − 3 𝑖𝑠 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Solution #7: 𝑟 = 6
𝑷 𝟔 = (𝟔) 𝟐
−𝟓 𝟔 + 𝟔
𝑃 6 = 36 − 30 + 6
𝑃 6 = 12
Since P(6)≠0, 𝑥 − 6 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Solution #8: 𝑟 = −6
𝑷 −𝟔 = (−𝟔) 𝟐
−𝟓 −𝟔 + 𝟔
𝑃 −6 = 36 + 30 + 6
𝑃 −6 = 72
Since P(-6)≠0, 𝑥 + 6 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.
Therefore the two factors of 𝒙 𝟐
− 𝟓𝒙 + 𝟔 = 𝟎 𝑎𝑟𝑒 ( 𝑥 −
Solution #4: 𝑟 = −2
𝑷 −𝟐 = (−𝟐) 𝟐
−𝟓 −𝟐 + 𝟔
𝑃 −2 = 4 + 10 + 6
𝑃 −2 = 20
Since P(-2)≠0, 𝑥 + 2 𝑖𝑠 𝑵𝑶𝑻 𝒂 𝒇𝒂𝒄𝒕𝒐𝒓.

9. HOW TO PLAY
This is a FACT-OR-BLUFF game played in five rounds. During each round, players
will solve a given problem. There should be one person who will stand as the
“MASTER BLUFFER or the MOTHER/FATHER FACTer”. He/she will tell
whether the answer of the players is a fact or a bluff.
3. Create a scorecard.
4. During rounds 1 and 2, players will
2. Have a deal with other players. Set the consequence/s which will be given to the
loser/s.
ROUND
1
ROUND
2
ROUND
3
ROUND
4
ROUND
5
TOTAL
SCORE
5. The person/s with the highest score will win.
TIP:
THERE’S REALLY NO SPECIAL STRATEGY INVOLVED. IT’S ALL ABOUT
YOUR UNDERSTANDING AND DETERMINATION TO WIN. THESE MAKE
THIS GAME FUN AND EXCITING. ENJOY!
1. Choose the “MASTER BLUFFER/MOTHER FACTer”.
get 1 pt. for each correct answer. Those players who will score 5pts. or below in the
first two rounds will be eliminated. In rounds 3 and 4, two points will be given for
each correct answer. Those players who will score 10pts. or below in the 3rd and 4th
round will not move on to the next round. For round 5, the players will be given a
chance to select 5 factors from the given binomials. However, if they obtain two or
more incorrect factors in this round, no points will be given.

10. A. REMAINDER, REMEMBER???
State whether the given remainder is correct or not. Choose
FACT, if the remainder is correct and BLUFF if it is
incorrect.
1. (𝑥4
−𝑥3
+ 2) ÷ (𝑥 + 2); 𝑅 = 0
2. (𝑥3
−2𝑥2
+ 𝑥 + 6) ÷ (𝑥 − 3); 𝑅 = 18
3. (𝑥4
−3𝑥3
+ 4𝑥2
− 6𝑥 + 4) ÷ (𝑥 − 2); 𝑅 = 0
4. (𝑥4
−2𝑥3
+ 2𝑥2
− 1) ÷ (𝑥 − 1); 𝑅 = 2
5. (3𝑥2
+5𝑥3
− 8𝑥 − 6) ÷ (𝑥 + 1); 𝑅 = 0

12. B. I REMAINDER!!!
Determine the remainder when 𝑃(𝑥) is divided by
the given binomial.
1. (𝑥2
−3𝑥 + 7) ÷ (𝑥 + 5)
2. (𝑥4−𝑥3 + 2) ÷ (𝑥 − 1)
3. (𝑥3
−𝑥2
− 8𝑥 + 12) ÷ (𝑥 − 2)
4. (8𝑥3
−4𝑥2
+ 2𝑥 − 1) ÷ (2𝑥 − 1)
5. (25𝑥2 − 10𝑥 − 8) ÷ (5𝑥 + 2)

14. A. WHAT’s THE FACTor?
Indicate whether the binomial is a factor of
the given polynomial.
1. 𝑃 𝑥 = 𝑥3 − 7𝑥 + 6
2. 𝑃 𝑥 = 2𝑥3 − 7𝑥 − 2
3. 𝑃 𝑥 = 4𝑥4
− 3𝑥3
− 𝑥2
+ 2𝑥 − 2
4. 𝑃 𝑥 = 𝑥4 + 3𝑥3 − 4𝑥2;
5. 𝑃 𝑥 = 16𝑥4 + 24𝑥3 + 4𝑥2 − 2𝑥 − 4;
(𝒙 + 𝟏)
(𝒙 + 𝟐)
(𝒙 − 𝟏)
(𝒙 − 𝟐)
(𝒙 + 𝟏) (𝒙 − 𝟏)
(𝒙 + 𝟒) (𝒙 − 𝟒)
(𝟐𝒙 − 𝟏) (𝟐𝒙 + 𝟏)

16. B. I Remainder the FACTor!
Use the Factor Theorem to identify the remainder and
determine whether or not the first polynomial is a factor of
the second.
1. 𝑥 − 1 ; 𝑥2 + 2𝑥 + 5
2. 𝑥 + 1 ; 𝑥4
− 𝑥 − 2
3. 𝑥 − 2 ; 2𝑥3 − 9𝑥2 + 9𝑥 + 2
4. 𝑎 − 1 ; 𝑎3 − 2𝑎2 + 𝑎 − 2
5. 𝑦 + 3 ; 2𝑦3 + 𝑦2 − 13𝑦 + 6

18. Find the 5 factors of the given polynomial.
𝑥5 − 3𝑥4 − 11𝑥3 + 27𝑥2 + 10𝑥 − 24
(𝑥 − 1)
(𝑥 + 1)
(𝑥 + 2)
(𝑥 − 2)
(𝑥 + 3)
(𝑥 − 3)
(𝑥 + 4)
(𝑥 − 4)
(𝑥 + 5)
(𝑥 − 5)
(𝑥 + 6)
(𝑥 − 6)
A. I FACTor!

19. BLUFF BOARD
1. BLUFF; R=26
2. FACT
3. FACT
4. BLUFF; R=0
5. FACT
ACTIVITYCARDNO. 1

20. BLUFF BOARD
1. R=47
2. R=2
3. R=8
4. R=0
5. R=0
ACTIVITYCARDNO. 2

21. BLUFF BOARD
1. (𝑥 − 1)
ASSESSMENTCARDNO. 1
2. (𝑥 − 2)
3. (𝑥 − 1)
4. (𝑥 + 4)
5. (2𝑥 − 1)

22. BLUFF BOARD
1. 𝑅 = 8; 𝑁𝑂𝑇 𝐴 𝐹𝐴𝐶𝑇𝑂𝑅
ASSESSMENTCARDNO. 2
2. 𝑅 = 0; 𝐹𝐴𝐶𝑇𝑂𝑅
3. 𝑅 = 0; 𝐹𝐴𝐶𝑇𝑂𝑅
4. 𝑅 = −2; 𝑁𝑂𝑇 𝐴 𝐹𝐴𝐶𝑇𝑂𝑅
5. 𝑅 = 0; 𝐹𝐴𝐶𝑇𝑂𝑅

23. BLUFF BOARD
ENRICHMENTCARD
(𝑥 − 1)
(𝑥 + 1)
(𝑥 − 2)
(𝑥 + 3)
(𝑥 − 4)

24. ENRICHMENTCARD
(𝑥 − 1)
(𝑥 + 1)
(𝑥 − 2)
(𝑥 + 3)
(𝑥 − 4)
REFERENCECARD
Mathematics – Grade 10
Learner’s Module
First Edition
https://www.brightstorm.com/math/algebra-
2/factoring/rational-roots-theorem/
Mathematics for the 21st Century Learner
Grade 10
DIWA TEXTBOOK

25. or
(FACTORS OF POLYNOMIAL)
ALL RIGHTS RESERVED 2019