𝒙𝟐 − 𝟏𝟔
(𝐱 + 𝟒)(𝐱 − 𝟒)
FACTORING POLYNOMIALS
Week 1, M8AL-1a-b-1
by: Mrs. Lalaine B. Sampan
MT I-Math
Sandayong Sur NHS
Factors completely different types of
polynomials
(polynomials with common monomial
factor, difference of two squares,
sum and difference of two cubes,
perfect square trinomials, and
general trinomials).
MONOMIAL
4(𝑎 + 𝑏)
BINOMIAL
𝑥 2 − 𝑦2
𝑥 3 ± 𝑦3
1
MGCQ
2
GCQ
3
TRINOMIAL
𝑥 2 + 2𝑥 + 1
𝑥 2 + 4𝑥 − 5
4 TERMS
𝑎𝑥 + 𝑎𝑦 + 𝑐𝑥 + 𝑐𝑦
MECQ
4
ECQ
5
LOCKDOWN
COMBINATION
2𝑥 2 − 18
2
5𝑥 + 10𝑥 + 5
In factoring polynomials by common monomial
factoring:
1. The first factor is determined by finding the
Greatest Common Factor (GCF) of the terms
in the polynomial.
2. The other factor is obtained by dividing each
term by the GCF.
FACTORS → 𝑷𝑹𝑶𝑫𝑼𝑪𝑻
1. 50 −4
2. −5 −16
= −200
= 80
3. 2 x + 3y
=
4. 3x 2x − 6y
= 6𝑥 2 −18𝑥𝑦
5. 2x 3 5x 4 + 3xy − 2
= 10𝑥 7 +6𝑥 4 𝑦 −4𝑥 3
2𝑥 +6𝑦
The factors of 12 are 1,2,3,4,6,12
The prime factorization of 12 is 𝟐𝟐 ∙ 𝟑
2 can have exponents 0,1,2 ; 3 factors
3 can have exponents 0 and 1; 2 factors
3(2)= 6 factors
Likewise, 15 has 4 factors i.e. 1,3,5,15
The prime factorization of 15 is 3∙ 𝟓
3 can have exponents 0 and 1; 2 factors
5 can have exponents 0 and 1; 2 factors
2(2)= 4 factors
𝟐𝟎 = 𝟏
𝟐𝟏 = 𝟐
𝟐𝟐 = 𝟒
𝟐 𝟑
𝟒 𝟑
𝟑𝟎 = 𝟏
𝟑𝟏 = 𝟑
=𝟔
= 𝟏𝟐
3 .
1. The GCF of 12 and 15 is _____
𝟑𝒙
2. The GCF of 12𝑥 and 15𝑥 is ______
.
𝟑𝒚 .
3. The GCF of 12𝑥𝑦 and 15𝑦 is ________
𝟐 𝟐
4 3
2 2
𝟑𝒙
𝒚
4. The GCF of 12𝑥 𝑦 and 15𝑥 𝑦 is _________.
This is how the GCF of no.4 is obtained:
1.The factored form of 12𝑥 2 𝑦 2 is 𝟑 ∙ 4 ∙ 𝒙𝟐 ∙ 𝒚𝟐
2. The factored form of 15𝑥 4 𝑦 3 is 𝟑 ∙ 5 ∙ 𝒙𝟐 ∙ 𝑥 2 ∙ 𝒚𝟐 ∙ 𝑦
Therefore, the GCF is 𝟑𝒙𝟐 𝒚𝟐
Notice that between 12𝑥 2 𝑦 2 and 15𝑥 4 𝑦 3 , the lowest
exponent of x is 2 and the lowest exponent of y is also 2.
To factor a number means to break it up into numbers that can be
multiplied together to get the original number. Factoring is
unmultiplying. It is the reverse process of finding the product.
1. 𝟐𝒂 + 𝟒𝒃 = 2 (𝑎 + 2𝑏)
Greatest Common Monomial Factor
This is how to factor, GCMF is 2 ;
2𝑎 + 4𝑏 2𝑎
4𝑏
=
+
= 𝑎 +2𝑏
2
2
2
2. 𝟖𝒙𝒚𝟐 – 𝟏𝟐𝒙𝟐 𝒚𝟑 = 4𝑥𝑦 2 (2 − 3𝑥𝑦)
Greatest Common Monomial Factor
Factoring out the GCMF,𝟒𝒙𝒚𝟐
8𝑥𝑦 2 − 12𝑥 2 𝑦 3
2 𝟐−𝟏
4𝑥𝑦
𝟎
𝒙
= 𝒙 𝒚𝟑−𝟐 = 𝒚
𝒙 =𝟏
8𝑥𝑦 2 12𝑥 2 𝑦 3
−
= 2 −3xy
2
2
4𝑥𝑦
4𝑥𝑦
In both terms, the
lowest exponent of x
is 1 and the lowest
exponent of y is 2.
In dividing powers with
the same base ,
subtract the exponents
In factoring polynomials by grouping:
1. Check if it has 4 terms and the terms have no
common factor.
2.Group pairs of terms so that the terms in each
group have a common factor.
3.Find the GCF in each group. If there is a common
binomial factor, then factor the polynomial.
4. Write the polynomial as a product of the
common factor and the other remaining factor.
Example 1: 2𝑦 + 2𝑥 + 𝑎𝑦 + 𝑎𝑥
+
+
Step 1:
+
2𝑦 + 2𝑥) 𝑎 (
𝑎𝑦
+
𝑎𝑥
)
(
𝑦
+
𝑥
+
𝑦
+
𝑥
Step 2: 2
2
𝑎
or
Step 3: (𝑦 + 𝑥) 2 + 𝑎
(a+2) 𝑥 + 𝑦
Example 1: 2𝑦 + 2𝑥 + 𝑎𝑦 + 𝑎𝑥
+
Step 1:
+
+
2𝑦
+
𝑎𝑦
2𝑥) + 𝑎𝑥
Step 2: 𝑦 ( 2 + 𝑎 ) + 𝑥(2 + 𝑎
𝑦 𝑦
𝑥
Step 3: ( 2 + 𝑎) + 𝑥 or
(a+2) 𝑥 + 𝑦
Alternative Solution
Example 2: 2𝑦 + 2𝑥 − 𝑎𝑦 − 𝑎𝑥
+
+
Step 1:
−1
2𝑦 + 2𝑥
𝑎𝑦
+
𝑎𝑥
)
(
)
𝑦
+
𝑥
(
− 𝑎 𝑦+𝑥
Step 2: 2
2
𝑎
Step 3: ( 𝑦 + 𝑥 ) 2 − 𝑎 or
(2 − a) 𝑥 + 𝑦
Example 2: 2𝑦 + 2𝑥 − 𝑎𝑦 − 𝑎𝑥
−
Step 1:
+
−
2𝑦 − 𝑎𝑦
2𝑥
−
𝑎𝑥
Step 2: 𝑦 ( 2 − 𝑎 ) + 𝑥(2 − 𝑎 )
𝑦
𝑥
Step 3: ( 2 − 𝑎) 𝑦 + 𝑥 or
(2 − 𝑎) 𝑥 + 𝑦
Alternative Solution
𝑥2 − 𝑦2 = 𝑥 + 𝑦 𝑥 − 𝑦
In factoring difference of two squares:
1. Get the principal square root of each
of the two squares. If not perfect squares ,check if
it has GCF.
2. Using the square roots, form two factors,
one a sum and the other a difference.
The factors are a sum & a
difference of two terms
1. x + 5 x − 5
=
𝑥 2 −25
2. 2x + 7 2x − 7 =
4𝑥 2 − 49
3.
1
x
4
+5
1
x
4
4.
3
x
2
1
3
3
x
2
+
1 2
𝑥 − 25
−5 =
16
−
1
3
9 2 1
𝑥 −
=
4
9
The product is a
difference
of two squares
(2𝑥 2 )2 −(5𝑦)2
4
2
4𝑥
−
25
𝑦
5. 2x + 5y 2x − 5y =
2
2
2
1. 9𝑥 − 4
2. 16𝑥 2 − 5
3. 81𝑥 2 − 64
4. 49𝑥 4 − 25𝑦 6
Factoring no.3
9 2
4
𝑥 −
7
49
4
5.
6. 27𝑥 − 9
7. 9𝑥 2 − 𝑦 2
4 2
9
8. 𝑥 −
9
25
PERFECT
SQUARES
𝟏𝟐 = 𝟏
𝟐𝟐 = 𝟒
𝟑𝟐 = 𝟗
𝟒𝟐 = 𝟏𝟔
𝟓𝟐 = 𝟐𝟓
𝟔𝟐 = 𝟑𝟔
𝟕𝟐 = 𝟒𝟗
𝟖𝟐 = 𝟔𝟒
𝟗𝟐 = 𝟖𝟏
𝟏𝟎𝟐 = 𝟏𝟎𝟎
81𝑥 2 = 9x
;
64 = 8
𝟖𝟏𝒙𝟐 − 𝟔𝟒 = ( + )(
− )
49𝑥 4 = 7𝑥 2
;
25𝑦 6 = 5𝑦 3
49𝒙𝟒 − 𝟐𝟓𝒚𝟔 = (𝟕𝒙𝟐 + 𝟓𝒚𝟑 )(𝟕𝒙𝟐 − 𝟓𝒚𝟑 )
Factoring no. 4
Difference of Two
Squares
1. 𝟏𝟑𝒙𝟐 − 𝟓𝟐 = 13 𝒙𝟐 − 𝟒
= 13 𝑥 + 2 𝑥 − 2
Not a
Greatest Common
difference of
Monomial Factor
two squares
2. 𝟐𝒙𝟑 − 𝟏𝟖𝒙 =2𝑥 𝑥 2 − 9
= 2𝑥 𝑥 + 3 𝑥 − 3
3. 𝟐𝟐 𝟏𝟖
=(20 + 2)(20 − 2)
𝑥 3 + 𝑦 3 = 𝑥 + 𝑦 𝑥 2 − 𝑥𝑦 + 𝑦 2
𝑥 3 − 𝑦 3 = 𝑥 − 𝑦 𝑥 2 + 𝑥𝑦 + 𝑦 2
1. Get the cube root of each term to find the first factor.
2. For the 2nd factor, square the first term of the first factor,
take the opposite sign of the product of the terms of
the first factor and lastly square the second term of the
first factor.
𝑥2 −
5𝑥 + 25
25)
−5𝑥
𝑥 + 5 (𝑥
by Box Method, the product is shown below.
3
𝑥
5𝑥 2
−5𝑥 2 25𝑥
−25𝑥 125
x + 5 (x 2 − 5x + 25) = 𝑥 3 + 125
sum of two cubes
PERFECT
CUBES
𝟏𝟑 = 𝟏
𝟐𝟑 = 𝟖
𝟑𝟑 = 𝟐𝟕
𝟒𝟑 = 𝟔𝟒
𝟓𝟑 = 𝟏𝟐𝟓
𝟔𝟑 = 𝟐𝟏𝟔
𝟕𝟑 = 𝟑𝟒𝟑
𝟖𝟑 = 𝟓𝟏𝟐
𝟗𝟑 = 𝟕𝟐𝟗
𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎
This is how the factored form of no.2 is obtained:
8𝑥 3 + 27 = (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9)
Terms
2
2
Terms
1. ( ) = 4𝑥
3
3
of the
1. 8x = 2𝑥
of the 2. −
6𝑥
−6𝑥
=
nd
3
2
st
1
2. 27 = 3
3. ( )2 = 9
factor
factor
PERFECT
CUBES
𝟏𝟑 = 𝟏
9
𝟐𝟑 = 𝟖
𝟑
For the 2nd factor: 𝟑𝟑 = 𝟐𝟕
𝟒 = 𝟔𝟒
S- square the 1st
𝟓𝟑 = 𝟏𝟐𝟓
term
𝟔𝟑 = 𝟐𝟏𝟔
O-opposite sign
𝟕𝟑 = 𝟑𝟒𝟑
for the product
𝟖𝟑 = 𝟓𝟏𝟐
of the 1st & the
𝟗𝟑 = 𝟕𝟐𝟗
2nd terms of
𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎
Factor 125𝑥 6 − 8𝑧
Terms of the 1st factor:
3
1. 125𝑥 6 = 5𝑥22
3
2. −8𝑧 9 = −2𝑧 3
Terms of the 2nd factor:
the 1st factor
A- always take the
4
2
1. (
) = 25𝑥
P- positive sign for
square of
2. −(
)(
) =10𝑥 2 𝑧 3 the
the last term
2 4𝑧 6
3. (
) =
Factored form:
+
(
)(
+
)
Factoring
Technique
Given
6𝑥 25− 4xy 4− 23x − 2y
Factored
2𝑥
𝑦
3𝑥
𝑦
Factored
2𝑥 3𝑥 −−2𝑦 Given
− 1(3𝑥
−−2𝑦)
=
2x
3y
4
4
Form
Form
𝑥 −𝑦 1)(3𝑥𝑥 −𝑦 2𝑦)
(2𝑥
2 𝑥2 −
2 𝑦 2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
𝑥
−
𝑦
= (𝑥 + 𝑦)(𝑥 − 𝟐𝑦)
GCF/
𝟔𝒙 − 𝟒𝒙𝒚 −
𝟒
𝟓
𝟒
𝟐
2
4 4𝑥;
5
10 = 7𝑦
−
16𝑥
9𝑥 8𝒚(𝟐𝒙
= 3𝑥=
; 𝟑𝒚)
49𝑦25
𝟑𝒙5+ 𝟐𝒚
By Grouping 𝟐𝒙 𝒚 − 𝟑𝒙 𝒚 𝒙
Difference
of Two
Squares
3 3+ 𝑦 3 3= (𝑥 + 𝑦)(𝑥 2 2− 𝑥𝑦 + 𝑦 2 )2
𝑥𝑥
− 𝑦 = (𝑥 − 𝑦)(𝑥 + 𝑥𝑦 + 𝑦 )
3
3
3
3
3
3
; 3436 =
; 2;
27𝑥27𝑥
= 3𝑥=; 3𝑥 −125𝑦
= 7−5𝑦
(𝟒𝒙
+
𝟓)
𝟐
2 2
𝟖 − 𝟒𝟗𝒚𝟏𝟎
2
𝟗𝒙
2
2
16𝒙 −(3𝑥)
𝟐𝟓
=
9𝑥
;
−
3𝑥
7
=
(3𝑥)= 9𝑥 ; − 3𝑥 −5𝑦 =
(𝟒𝒙722−
𝟓) 4
2
2
−
21𝑥;
=
49
15𝑥𝑦 ; (5𝑦 ) = 25𝑦
3𝑥(𝑥
3)(𝑥2 −
− 𝑥𝑦
3) + 𝑦 2 )
Sum and
6(𝑥23−9)
+𝑦 3 )==3𝑥(𝑥
6(𝑥 ++𝑦)(𝑥
(𝟑𝒙 + 𝟕)
𝟑 − 𝟏𝟐𝟓𝒚𝟔
Difference 𝟐𝟕𝒙𝟑 + 𝟑𝟒𝟑
𝟐𝟕𝒙
(𝟗𝒙𝟐 − 𝟐𝟏𝒙 + 𝟒𝟗)
of Two Cubes
Complete
𝟑
𝟑𝒙
−
𝟐𝟕𝒙
Factorization
𝟑𝒙 𝒙 + 𝟑
(𝐱 − 𝟑)
𝟑
𝟑
𝟔𝒙 + 𝟔𝒚
𝟐𝒙 − 𝟏
(𝟑𝒙 − 𝟐𝒚)
(𝟑𝒙𝟒 + 𝟕𝒚𝟓 )
(𝟑𝒙𝟒 − 𝟕𝒚𝟓 )
(𝟑𝒙-5𝒚𝟐 )
(𝟗𝒙𝟐 + 𝟏𝟓𝒙𝒚𝟐 + 𝟐𝟓𝒚𝟒)
6 𝒙+𝒚
(𝒙𝟐 − 𝒙𝒚 + 𝒚𝟐 )
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Learn from it .
Live with it.
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Learner's Material in Math 8.
First Quarter Enhanced Detailed Lesson Plans
in Math 8 .(2018). Danao City.
Workbook in Math 8. (2018). Danao City.