factoring polynomials using different methods

1. Mathematics
Factoring
Polynomials
Presented by

2. • A mathematical expression consisting of variables, coefficients,
and exponents, combined using addition, subtraction, and
multiplication.
What is Polynomials?
Progress:
𝑎𝑥𝑛
+ 𝑏𝑥𝑛−1
+ 𝑐𝑥 + 𝑑
• Coefficients (a,b,c,d), are numerical factors that multiply
the variables.
• Variables (x), represent unknown values.
• Exponents (n), denote the power to which the variable is
raised

3. Greatest Common
Monomial Factor
01
Is the reverse of multiplying a
monomial by a polynomial
Sum or Difference
of two cubes
03
Multiplying polynomials that
result in a sum or a difference
of two cubes
Difference of two
Squares
02 When you have the subtraction
of two perfect squares
Factoring Polynomials Techniques

4. The largest expression that divides each term of the polynomial evenly. It can be a
numerical factor, a variable, or a combination of both.
Note: Factoring by Greatest Common Monomial Factor is the
reverse of multiplying a monomial by a polynomial. Progress:
A. Greatest Common Monomial Factor
General equation:
(a)(b+c) = ab + ac
Reverse:
ab+ac = a(b+c)
For examples:
1. 8x + 16
2. -4x – 8xy
Steps:
S1. Get the prime factors of the numerical coefficient/Literal coefficients.
S2. Divide the original expression by the GCMF
S3. Write the factored form of the expression

5. The largest expression that divides each term of the polynomial evenly. It can be a
numerical factor, a variable, or a combination of both.
Note: Factoring by Greatest Common Monomial Factor is the
reverse of multiplying a monomial by a polynomial. Progress:
A. Greatest Common Monomial Factor
For examples:
1. 8x + 16
Ans. 8(x+2)
2. -4x – 8xy
Ans. (-4x)(1+2y)
Steps:
S1. Get the prime factors of the numerical coefficient/Literal coefficients.
S2. Divide the original expression by the GCMF
S3. Write the factored form of the expression
General equation:
(a)(b+c) = ab + ac
Reverse:
ab+ac = a(b+c)

6. The difference of two squares is a special factoring pattern that occurs when you have a
square term subtracted by another square term.
Note: You can only use this technique if and only if all the following, are satisfied:
a. It is binomial, and the terms are perfect squares
b. It is a difference of the two terms.
Progress:
B. Difference of Two Square
For examples:
1. 𝑎4
− 36
Ans. (𝑎2
− 6)(𝑎2
+ 6)
2. 16𝑥2
− 25𝑏2
Ans. (4x—5b)(4x+5b)
Steps:
S1. Find the square root of each term
S2. Follow the pattern: 𝑎2 − 𝑏2= (a-b)(a+b)
General form:
(a-b)(a+b)=(𝑎2
− 𝑏2
)
Reverse:
𝑎2
− 𝑏2
= (a-b)(a+b)

7. Another special factoring pattern. This occurs when you have a cube term added or
subtracted to another cube term.
Note: You can only use this technique if and only if all the following, are satisfied:
a. It is binomial, and the terms are perfect cubes
b. It is a sum or difference of the two terms.
Progress:
C. Sum or Difference of Two Cubes
For examples:
1. 𝑥3
+ 8
Ans. (x+2)(𝑥2
− 2𝑥 + 4)
2. 8𝑥3
− 27𝑦3
Ans. (2x-3y)(4𝑥2
+ 6𝑥𝑦 + 9𝑦),
Steps:
S1. Find the cube root of the factors.
S2. Follow the pattern: 𝑎3
− 𝑏3
=(a-b)(𝑎2
+ 𝑎𝑏 + 𝑏2
),
𝑎3
+ 𝑏3
=(a+b)(𝑎2
− 𝑎𝑏 + 𝑏2
)
General form:
(a-b)(𝒂𝟐
+ 𝒂𝒃 + 𝒃𝟐
) = 𝒂𝟑
− 𝒃𝟑
(a+b)(𝒂𝟐
− 𝒂𝒃 + 𝒃𝟐
) = 𝒂𝟑
− 𝒃𝟑
Reverse:
𝒂𝟑
− 𝒃𝟑
=(a-b)(𝒂𝟐
+ 𝒂𝒃 + 𝒃𝟐
)
𝒂𝟑
+ 𝒃𝟑
=(a+b)(𝒂𝟐
− 𝒂𝒃 + 𝒃𝟐
)

8. Factoring

9. Perfet Square
Trinomial 01 General Trinomials
02
Factoring Techniques

10. A Trinomial is a perfect square if the first and the least terms are perfect squares, and the
middle term is twice the product of the square root of the first term and the square root of
the last term.
Progress:
A. Perfect Square Trinomial
For examples:
1. 𝑥2
− 6𝑥 + 9
Ans. (𝒙 − 𝟑)𝟐
1. 4𝑥4
+ 36𝑥2
+81
Ans. (𝟐𝑥2
+ 𝟗)𝟐
General form:
(a+b)(a+b)=𝒂𝟐
+ 𝒂𝒃 + 𝒂𝒃 + 𝒃𝟐
= 𝒂𝟐
+ 𝟐𝒂𝒃 + 𝒃𝟐
(a-b)(a-b)=𝒂𝟐
− 𝒂𝒃 − 𝒂𝒃 + 𝒃𝟐
= 𝒂𝟐
− 𝟐𝒂𝒃 + 𝒃𝟐
Reverse:
𝒂𝟐
+ 𝟐𝒂𝒃 + 𝒃𝟐
= (𝒂 + 𝒃)𝟐
or (a+b)(a+b)
𝒂𝟐
− 𝟐𝒂𝒃 + 𝒃𝟐
= (𝒂 − 𝒃)𝟐
or (a-b)(a-b)
Steps:
S1: Square root the first and last term.
S2: Twice the product of the square root of the first and last term
S3: Substitute the equation

11. Use this factoring techniques whenever the polynomial is a trinomial and it is not a perfect
square.
Progress:
B. General Trinomial
For examples:
1. 𝑥2
+ 7𝑥 + 12
Ans. (x+3)(x+4)
1. 𝑥2
− 6𝑥+8
Ans. (x-4)(x-2)
General form:
(x+a)(x+b)=𝒙𝟐
+ 𝒃𝒙 + 𝒂𝒙 + 𝒂𝒃
= 𝒙𝟐
+ 𝒂 + 𝒃 𝒙 + 𝒂𝒃
Reverse:
𝒙𝟐
+ 𝒂 + 𝒃 𝒙 + 𝒂𝒃 = (x+a)(x+b)
Note:
 Recall Multiplication of Binomial
 You can use FOIl Method to check your answer
Steps:
S1. Find the two numbers that when multiplied, the product is the
constant number or third term and the sum is the numerical
coefficient of the second term.
S2: Add this product to the variable to form two binomial factors

12. Welcome to our Math Mystery Escape
Room!! Here, you will find that Venus has a
beautiful name and is the second planet from
the Sun. It’s terribly hot, even hotter than
Mercury, and its atmosphere is extremely
poisonous
About the introduction
Progress:

13. She is a very funny Turkish girl. She
enjoys reading, playing with her
dogs and visiting her partner every
weekend. She is specially good in
psychology and mathematics as she
has a very analytical mind. Her
friends describe her as loyal and
loving
This is ana
Progress:

14. We don’t know where he is from
Progress:
This is finnick!
He is 15 years old
Finnick loves chess and parkour
He does not have lots of friends: only 1
He has a tortoise called Edison
He hates math but is good at them
Finnick has always a smile on his face

15. This is robin
Progress:
Special traits
Tall, no freckles, smiley and fun
nationality
Unknown
hobbies
Soccer, rugby and football

16. gathering the data
Progress:
80
60
40
20
0
Ana Robin Finnick
To modify this graph, click on it, follow the link, change the data with your own and replace the graph in this
slide
Her strong: be punctual
Ana “O’clock” is an excellent organizer
Books and knowledge
Robin loves to read, almost as much as
Ana
Math
Finnick hates math although he is good at it

17. Understa
nding it
02 Here you can describe the
content of the section

18. utilities for deciphering this math
mystery
Progress:
SOME
RANDOM
NOTES
It is important to have
somewhere to write
Talk to my
teacher
Your math teachers
are the best! They
know it all
HANDY
NOTEBOOK
Afterwards, put all your
notes in the same
place
A PRIZE TO
WIN!
Will you be the first to
decipher the mystery?

19. Progress:
THERE CAN BE
TWO PATHS...
BUT ONLY ONE
HAS THE
SOLUTION
It’s OK if you have different paths
for solving a math problem.
Hooray: You’re actually on the
right track!
It may come to you just in the
verge of giving up, so keep trying.
You will find the answer if you are
prepared!
Are you STUCK?

20. Mathematics is not about
numbers, equations,
computations or algorithms:
it is about understanding.
—SOMEONE FAMOUS

21. Collect
ing
eviden
ce
03
Here you can describe the
content of the section

22. REFEREN
CE
Ceres is located in
the asteroid belt
Evidence found
ANCIENT
NOTES
Pluto is considered
a dwarf planet
EVIDENCE
Earth is the planet
where we live on
MODERN
NOTES
Saturn is a gas
giant and has rings
Progress:

23. Let’s see
what we
have
found...

24. Solut
ion
found
!
Here you can describe the
content of the section
04

25. Have you found the solution on your
computer?
I have not! However, you can
actually change the image on the
screen with your own by
replacing it and uploading a
photo from your own computer!
Awesome!
Progress:

26. No way! However, you can
actually change the image on the
screen with your own by
replacing it and uploading a
photo from your own computer!
Awesome!
Or, maybe, on your tablet?
Progress:

27. Hooray! I’m almost there! Bear in
mind that you can change the
image on the screen with your
own by replacing it and uploading
a photo from your own computer!
Nope, it’s on my phone! Let’s see
what it is!
Progress:

28. Ana is 11, robin 22
and finnick 15
Yay! And Finnick is from Finland!

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