Polynomials are algebraic expressions that are consist of variables and coefficients. We can perform arithmetic operations such as subtraction, addition, multiplication and division. This presentation is all about factoring completely different types of polynomials. There four types of polynomials to factor that would be discuss in this presentation

1. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Polynomials
Polynomials are algebraic
expressions that consist of
variables and coefficients.
Variables are also sometimes
called indeterminate. We can
perform arithmetic operations
such as addition, subtraction
multiplication and also positive
integer exponents for
polynomial expressions but
not division by variable.

2. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Examples of Polynomials:
𝟔𝟒𝒛 𝟑
− 𝟖𝒚 𝟑
𝟏𝟐𝒙 𝟑
𝒚 𝟓
− 𝟐𝟎𝒙 𝟓
𝒚 𝟐
𝒛
𝟐𝟏𝐚𝐛 – 𝟑𝟓𝐛𝐜 + 𝟕𝐛 𝟐
𝐜 𝟑
𝟏𝟖𝒉 𝟐
+ 𝟏𝟐𝐡 + 𝟐

3. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
The Different Types of
Polynomials to Factor
• Polynomials that are
different of two squares
• Polynomials that are
sum and difference of
two cubes
• Polynomials with
common monomial
factor
• Polynomials that are
perfect square trinomials

4. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Polynomials with Common Monomial Factor
To factor completely the polynomial with common
monomial factor, here are the steps for you to remember:
a. Find the greatest common factor of the numerical
coefficients.
b. Find the variable with the least exponent that appears in
each term of the polynomial.
c. The product of the greatest common factor in (a) and (b) is
the GCF of the polynomial.
d. To completely factor the given polynomial, divide the
polynomial by its GCF, the result in quotient is the other
factor.

5. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #1: Polynomials with Common Monomial
Factor
Factor: 8𝒂 𝟐
+ 𝟔𝒂
a. The GCF of 8 and 6 is 2.
b. a is common to all terms and 1 is the smallest
exponent for a, thus a is the GCF of the
variables.
c. (2)(a); hence, 2a is the common factor of the
variable.
d. 8𝒂 𝟐
+ 𝟔𝒂 = 4a+3 the quotient
2a
Thus, the factored form of 8𝒂 𝟐
+ 𝟔𝒂 is 2a(4a+3).

6. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #2: Polynomials with Common Monomial
Factor
Factor: 𝟏𝟐𝒙 𝟑 𝒚 𝟓 − 𝟐𝟎𝒙 𝟓 𝒚 𝟐 𝒛.
𝟒 𝟑 𝒙 𝟑
𝒚 𝟐
𝒚 𝟑
𝟒 𝟓 𝒙 𝟑
𝒙 𝟐
𝒚 𝟐
𝒚 𝟑
𝒛
𝟒𝒙 𝟑 𝒚 𝟐 𝟑 𝒚 𝟑 − 𝟓 𝒙 𝟐 𝒚 𝟑 𝒛
𝟒𝒙 𝟑
𝒚 𝟐
𝟑𝒚 𝟑
− 𝟓𝒙 𝟐
𝒛
FACTORED FORM:
SOLUTION:

7. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #3: Polynomials with Common Monomial
Factor
Factor: 𝟐𝟏𝐚𝐛 – 𝟑𝟓𝐛𝐜 + 𝟕𝐛 𝟐
𝐜 𝟑
𝟕(𝟑)𝐚𝐛 – 𝟕(𝟓)𝐛𝐜 + 𝟕(𝒃)(𝒃) 𝐜 𝟑
𝟕𝐛(𝟑𝐚 − 𝟓𝐜 + 𝐛𝐜 𝟑
)
SOLUTION:
FACTORED FORM: 𝟕𝐛(𝟑𝐚 − 𝟓𝐜 + 𝐛𝐜 𝟑)

8. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Polynomials that are Difference of Two Squares
Remember the factored form of a polynomial that is a
difference of two squares is the sum and difference of the
square roots of the first and last term.
Here are the guides in factoring the difference of two squares:
a. Find the square root of each terms.
b. Write the product of the sum and difference of the square
roots.

9. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #1: Polynomials that are Difference of Two
Squares
Factor: 𝒙 𝟐
− 𝟒
a. 𝒙 𝟐 = 𝐱 and 4 = 2
b. (x-2)(x+2)
FACTORED FORM: (x-2)(x+2)

10. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #2: Polynomials that are Difference of Two
Squares
Factor: 𝟑𝒙𝒚 𝟐
− 𝟏𝟐𝒙𝒛 𝟐
a. Find the GCMF of the polynomial since there is
common in each term and then factor.
𝟑𝒙𝒚 𝟐
− 𝟏𝟐𝒙𝒛 𝟐
= 3x(𝒚 𝟐
− 𝟒𝒛 𝟐
)
b. 𝒚 𝟐 = y and 𝟒𝒛 𝟐 = 2z
c. 3x(y −2z)(y+2z)
FACTORED FORM: 3x(y −2z)(y+2z)

11. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #3: Polynomials that are Difference of Two
Squares
Factor: 𝟖𝟏𝒗 𝟒 − 𝟐𝟓𝒚 𝟒
a. 𝟖𝟏𝒗 𝟒 = 𝟗𝒗 𝟐
and 𝟐𝟓𝒚 𝟒 = 5y
b. (𝟗𝒗 𝟐
+ 5y)(𝟗𝒗 𝟐
− 5y)
FACTORED FORM: (𝟗𝒗 𝟐
+ 5y)(𝟗𝒗 𝟐
− 5y)

12. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Polynomials that are Sum and Difference of Two
Cubes
The pattern in factoring the sum and difference of two cubes:
a. 𝒙 𝟑
+ 𝒚 𝟑
= (x + y)(𝒙 𝟐
− 𝒙𝒚 + 𝒚 𝟐
)
b. 𝒙 𝟑
− 𝒚 𝟑
= (x − y)(𝒙 𝟐
+ 𝒙𝒚 + 𝒚 𝟐
)
Here is the guide in factoring the sum and difference of two cubes:
1. Find the cube root of the first and last terms.
2. If Sum of Two Cubes, write the sum of the cube root as the first
factor. If Difference of Two Cubes, write the difference of the cube
roots of the first and last terms as first factor.
3. For the second factor, get the trinomial factor by:
a. Squaring the first term of the first factor.
b. If Sum of two cubes, subtracting the product of the first and
second terms of the first factor, if difference of two cubes,
adding the product of the first and second terms of the first
factor.
c. Squaring the last term of the first factor.
4. Write them in factored form.

13. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials
Example #1: Polynomials that are Sum and
Difference of Two Cubes
Sum of Two Cubes:
Factor: 𝒙 𝟑 + 𝟖𝒚 𝟑
1. 𝒙 𝟑𝟑
= x and 𝟖𝒚 𝟑𝟑
= 2y
2. (x+2y)
3. a. (𝒙) 𝟐
= 𝒙 𝟐
b. 𝒙 − 2y = -2xy
c. (𝟐𝒚) 𝟐 = 𝟒𝒚 𝟐
4. (𝒙 + 𝟐𝒚)(𝒙 𝟐
− 2xy + 𝟒𝒚 𝟐
)
FACTORED FORM: (𝒙 + 𝟐𝒚)(𝒙 𝟐
− 2xy + 𝟒𝒚 𝟐
)

14. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #2: Polynomials that are Sum and
Difference of Two Cubes
Sum of Two Cubes:
Factor: 𝒎 𝟗 𝒏 + 𝟕𝟐𝟗𝒅 𝟑
= (𝒎 𝟑
𝒏 + 𝟗𝒅)((𝒎 𝟑
𝒏) 𝟐
− 𝒎 𝟑
𝒏 𝟗𝐝 + (𝟗𝒅) 𝟐
)
= (𝒎 𝟑 𝒏 + 𝟗𝒅) 𝒎 𝟔 𝒏 𝟐 − 𝟗𝒎 𝟑 𝒏𝐝 + 𝟖𝟏𝒅 𝟐
FACTORED FORM:
(𝒎 𝟑
𝒏 + 𝟗𝒅) 𝒎 𝟔
𝒏 𝟐
− 𝟗𝒎 𝟑
𝒏𝐝 + 𝟖𝟏𝒅 𝟐

15. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #3: Polynomials that are Sum and
Difference of Two Cubes
Difference of Two Cubes:
Factor: 𝒅 𝟑 + 𝒑 𝟑
1. 𝒅 𝟑𝟑
= d and 𝒑 𝟑𝟑
2. (d − p)
3. a. (𝒅) 𝟐
= 𝒅 𝟐
b. d + p = 𝒅𝒑
c. (𝒑) 𝟐
= 𝒑 𝟐
4. (d − p)(𝒅 𝟐
+ 𝒅𝒑+ 𝒑 𝟐
)
FACTORED FORM: (d − p)(𝒅 𝟐
+ 𝒅𝒑+ 𝒑 𝟐
)

16. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #4: Polynomials that are Sum and
Difference of Two Cubes
Difference of Two Cubes:
Factor: 𝟔𝟒𝒛 𝟑 − 𝟖𝒚 𝟑
= (𝟒𝒛 𝟑
𝒏 − 𝟐𝒚)((𝟒𝒛) 𝟐
− 𝟒𝒛 𝟐𝒚 + (𝟐𝒚) 𝟐
)
=(𝟒𝒛 𝟑 𝒏 − 𝟐𝒚) 𝟏𝟔𝒛 𝟐 − 𝟖𝐳𝐲 + 𝟒𝒚 𝟐
FACTORED FORM:
(𝟒𝒛 𝟑 𝒏 − 𝟐𝒚) 𝟏𝟔𝒛 𝟐 − 𝟖𝐳𝐲 + 𝟒𝒚 𝟐

17. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Polynomials that are Perfect Square Trinomials
To factor perfect square trinomials:
1. Get the square root of the first and last terms.
2. List down the square root as sum/difference of two terms as
the case may be.
You can use the following relationships to factor perfect square
trinomials:
(First term) 𝟐
+ 2 (First term)(Last term) + (Last term) 𝟐
= (First term+Last term) 𝟐
(First term) 𝟐
− 2 (First term)(Last term) + (Last term) 𝟐
= (First term − Last term) 𝟐

18. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #1: Polynomials that are Perfect Square
Trinomials
Factor: 𝟒𝒓 𝟐
− 𝟏𝟐𝐫 + 𝟗
a. Since 𝟒𝒓 𝟐
= (𝟐𝒓) 𝟐
and 𝟗 = (𝟑) 𝟐
, and since
𝟏𝟐𝐫 = 𝟐 𝟐𝐫 𝟑 then it follows the given
expression is a perfect square trinomial.
b. The square root of the first term is 𝟐𝐫 and the
square root of the last term is 𝟑 so that its
factored form is (𝟐𝐫 − 𝟑) 𝟐
following the pattern
shown in the last slide.
FACTORED FORM: (𝟐𝐫 − 𝟑) 𝟐

19. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #2: Polynomials that are Perfect Square
Trinomials
Factor: 75𝒕 𝟐
+ 𝟑𝟎𝐭 + 𝟑𝐭
a. Observe that 𝟑𝐭 is common to all terms, thus,
factoring it out first we have: 𝟑𝐭 𝟐𝟓𝒕 𝟐
+ 𝟏𝟎𝒕 + 𝟏
b. Notice that 𝟐𝟓𝒕 𝟐 = (𝟓𝒕) 𝟐 and 𝟏 = (𝟏) 𝟐, and
𝟏𝟎𝒕 = 𝟐 (𝟓𝒕)(𝟏), then 𝟐𝟓𝐭 + 𝟏𝟎 + 𝟏 is a perfect
square trinomial.
c. Factoring 𝟐𝟓𝒕 𝟐 + 𝟏𝟎𝒕 + 𝟏 is (𝟓𝒕 + 𝟏) 𝟐, thus, the
factors of the given expression is 𝟑𝐭(𝟓𝒕 + 𝟏) 𝟐
.
FACTORED FORM: 𝟑𝐭(𝟓𝒕 + 𝟏) 𝟐.

20. FACTOR COMPLETELY DIFFERENT TYPES OF POLYNOMIALS
Polynomials With
Common Monomial
Factor
Polynomials that are
Difference of Two
Squares
Polynomials that are
Sum and Difference of
Two Cubes
Polynomials that are
Perfect Square
Trinomials
Polynomials Example #3: Polynomials that are Perfect Square
Trinomials
Factor: 𝟏𝟖𝒉 𝟐
+ 𝟏𝟐𝐡 + 𝟐
= 𝟐 𝟗 𝒉 𝟐
+ 𝟐 𝟔 𝒉 + 𝟐
= 𝟐 𝟗𝒉 𝟐
+ 𝟔𝒉 + 𝟏
= 𝟐 𝟑𝒉 + 𝟏 𝟑𝒉 + 𝟏
= 𝟐(𝟑𝒉 + 𝟏) 𝟐
FACTORED FORM: 𝟐(𝟑𝒉 + 𝟏) 𝟐