Grade 08 Mathematics: Factoring Polynomials with Common Monomial Factor

LESSON 001 GRADE 08 MATHEMATICS: ALGEBRA

Factoring Polynomials with Common Monomial Factor 85.87 cm

How can I factor polynomials with common monomial factor?

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

A.Unlocking Key Concepts

COMMON MONOMIAL FACTOR A.COMMON

common means “shared” between two different things.

Anong meron sa isa, na meron sa isa?

B.MONOMIAL

A polynomial with ONLY one term.

Kapag yung polynomial ay walang addition or subtraction operation.

C.FACTOR

85.87 cm

A number or an algebraic expression that divides another number or expression evenly.

Ano yung mga number/s or expression na pwedeng i-divide sa isang numero or expression?

GENERAL DEFINITION

TRY THESE!

a. 54 and 99

b. 32 and 64

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c. 30 and 100

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

A.Unlocking Key Concepts ANSWERS:

TRY THESE!

a. 54 and 99 54: 1 2 3 6 9 18 27 57 GCF: 9 99: 1 3 9 11 33 99 b. 32 and 64 32: 1 2 4 8 16 32 GCF: 32 64: 1 2 4 8 18 32 64 c. 30 and 100 30: 1 2 3 5 6 10 15 30 GCF: 10 100: 1 2 4 5 10 20 25 50 100

REMEMBER HOW TO FIND THE GCF? (Example: 65 & 25)

Step 1: Write down all the Positive Factors of a given constant/ number.

Everything should be a whole number.

65: 1 5 13 65 25: 1 5 25 Step 2: Determine the Greatest (Pinakamataas) Common (Pareho) Factor between the two constants/numbers.

65: 1 5 13 65 25: 1 5 25 GCF: 5

GENERAL PROCEDURES

STEP 1: STEP 2: STEP 3: STEP 4:

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

A.Unlocking Key Concepts TRY THESE!

EXAMPLES

STEP 1

STEP 2 Expanding each Variable

STEP 3 Determining Common Factors

TERM

Solving the GCF

2 25x

1, 5, 25

x•x

x•x

3 55x

1, 5, 11, 55

x•x•x

x•x•x

60x

1, 2, 3, 4, 5, 12, 15, 20, 30, 60

x

x

STEP 2 Expanding each Variable

STEP 3 Determining Common Factors

TERM

STEP 1 Solving the GCF

5 x y 4 3 18x y

1

STEP 4 Common Monomial Factor

5x

STEP 4 Common Monomial Factor

x•x•x•x•x• x•x•x•x•x• y y

x•x•x•x•y• x•x•x•x•y• 1, 2, 3, 6, 9, 18 y•y y•y

2 2 14x y

1, 2, 7, 14

x•x•y•y

x•x•y•y

5 10x

1, 2, 5, 10

x•x•x•x•x

x•x•x•x•x

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2 2 1x or x

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

A.Unlocking Key Concepts TRY THESE!

EXAMPLES

TERM

STEP 1 Solving the GCF

STEP 2 Expanding each Variable

STEP 3 Determining Common Factors

1, 3, 5, 15, 25, 75

x•x•x•x•z• x•x•x•x•z• y•y•y y•y•y

2 5 2 25x z y

1, 5, 25

x•x•z•z•z• x•x•z•z•z• z•z•y•y z•z•y•y

2 4 4 50x z y

1, 2, 5, 10, 25, 50

x•x•z•z•z• x•x•z•z•z• z•y•y•y•y z•y•y•y•y

4 3 75x zy

STEP 1

STEP 2 Expanding each Variable

STEP 3 Determining Common Factors

TERM

Solving the GCF

2 2 15s t v

1, 3, 5, 15

s•s•t•t•v

s•s•t•t•v

4 45s t

1, 3, 5, 9, 15, 45

s•s•s•s•t

s•s•s•s•t

STEP 4 Common Monomial Factor

2 2 25x zy

STEP 4 Common Monomial Factor

1 8v

1, 2, 4, 8

v

v

2 2 19s t

1, 19

s•s•t•t

s•s•t•t

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

B.Introducing New Concepts NEW CONCEPTS

What is Factoring?

2 x +5x+6

FACTORING

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(x+3)(x+2)

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

B.Introducing New Concepts What are the Importance of Factoring?

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

B.Introducing New Concepts Factoring Polynomials with Common Monomial Factor

REMEMBER

EXAMPLES

STEPS ON FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTORS

Step 1: Determine the Common Monomial Factors on all the terms of the given Polynomial. Write the CMF in a Parenthesis. This will be the First Factor of the Polynomial.

S T E P 1

Step 2: For the Second Factor, this involves Division of Polynomials.

Divide each term to the CMF.

TERM

Solving the GCF

Expanding each Variable

Determining Common Factors

2x2

1, 2

x • x

x • x

12xy

1, 2, 3, 4, 6, 12

x • y

x • y

6x

1, 2, 3, 6

x

x

Common Monomial Factor

Step 3: Write down all the quotients in another parenthesis beside the first factor. This will be considered as the second factor.

TERM

S T E P 2

Dividing Each Term with CMF

Quotient

2

2 2𝑥

2𝑥 2x

𝑥

12𝑥𝑦

−12𝑥𝑦 2𝑥

−6𝑦

6x

6𝑥 2𝑥

3

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S T 2 E = 2𝑥 − 12𝑥𝑦 + 6𝑥 P = (2𝑥)(𝑥 − 6𝑦 + 3) 3

2x

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

B.Introducing New Concepts TRY THESE!

EXAMPLES 2 2

35𝑎 𝑏 − 5𝑎𝑏

3

6𝑤 − 10𝑤 + 2𝑤 4 3 = 6𝑤 − 10𝑤 + 2𝑤 = 2𝑤( ) 4 6𝑤 2𝑤 3 = = 3𝑤 = =1 2𝑤 2𝑤 3 −10𝑤 2 = = −5𝑤 2𝑤 3 2 = 2𝑤(3𝑤 − 5𝑤 + 1) 2

−3𝑝 + 15𝑝 + 6𝑝 4

2

3

3

= 12𝑎 𝑏 𝑐 𝑑 + 8𝑎𝑏𝑑 2 = (4𝑎𝑏𝑑 )( ) 2 3 3 2 12𝑎 𝑏 𝑐 𝑑 2 3 = = 3𝑎𝑏 𝑐 2 4𝑎𝑏𝑑 3 8𝑎𝑏𝑑 = = 2𝑑 2 4𝑎𝑏𝑑 2 2 3 = (4𝑎𝑏𝑑 )(3𝑎𝑏 𝑐 + 2𝑑)

= 35𝑎 𝑏 − 5𝑎𝑏 = (5𝑎𝑏)( ) 2 2 35𝑎 𝑏 = = 7𝑎𝑏 5𝑎𝑏 −5𝑎𝑏 = = −1 5𝑎𝑏 = (5𝑎𝑏)(7𝑎𝑏 − 1)

4

12𝑎 𝑏 𝑐 𝑑 + 8𝑎𝑏𝑑 2 3 3 2

2 2

4

2 3 3 2

= −3𝑝 + 15𝑝 + 6𝑝 = (3𝑝)( ) 4 6𝑝 −3𝑝 3 = =2 = = −𝑝 3𝑝 3𝑝 2 15𝑝 = = 5𝑝 3𝑝 3 = (3𝑝)(−𝑝 + 5𝑝 + 2)

5

4

2

8𝑟 − 20𝑟 − 12𝑟 5 4 2 = 8𝑟 − 20𝑟 − 12𝑟 2 = (2𝑟 )( ) 5 2 8𝑟 −12𝑟 3 = 2 = 4𝑟 = = −6 2 2𝑟 2𝑟 4 −20𝑟 2 = = −10𝑟 2 2𝑟 2 3 2 = (2𝑟 )(4𝑟 − 10𝑟 − 6) 3

2 2

5ℎ 𝑘 − 20ℎ 𝑘 + 100ℎ𝑘 3 2 2 = 5ℎ 𝑘 − 20ℎ 𝑘 + 100ℎ𝑘 = 5ℎ𝑘( ) 3 100ℎ𝑘 5ℎ 𝑘 2 = = 20 = =ℎ 5ℎ𝑘 5ℎ𝑘 2 2 −20ℎ 𝑘 = = −4ℎ𝑘 5ℎ𝑘 2 = 5ℎ𝑘(ℎ − 4ℎ𝑘 + 20)

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

C. More Examples TRY THESE!

ACTIVITY Given 1.

2.

3.

Factors

Given

4 3 16𝑝 + 4𝑝

11.

2 3 20 − 35𝑛 − 20𝑛

9𝑥 + 36

12.

6 3 2 9𝑥 − 63𝑥 − 90𝑥

63 + 45𝑏

13.

3 2 −3𝑘 + 15𝑘 − 6𝑘

3 2 50𝑝 + 50𝑝 − 20

4.

3

6𝑛 − 3𝑛

5

14.

5.

5

−6𝑟 − 6𝑟

4

15.

6.

12 6 63𝑥 − 35𝑥

16.

−90𝑥 + 100𝑥 + 60

7.

2 3 14𝑎 + 21𝑎 + 21𝑎

17.

2 3𝑚 + 9𝑚 + 27

8.

3 2 10𝑛 − 9𝑛 + 𝑛

18.

2 12𝑟 + 4𝑟 − 12

9.

2 −28𝑣 − 8𝑣 − 36

19.

2 64 + 40𝑥 + 72𝑥

10.

7 6 5 −8𝑥 + 24𝑥 + 12𝑥

20.

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3

32𝑛 + 28𝑛 − 20

5

2

−18𝑛 + 15𝑛 − 15

Factors

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Common Monomial Factor

C. More Examples TRY THESE!

ANSWER KEY Given 1.

2.

3.

Factors

4 3 16𝑝 + 4𝑝

63 + 45𝑏

𝟒𝒑 + 𝟏

11.

(𝟗)(𝒙 + 𝟒)

12.

6 3 2 9𝑥 − 63𝑥 − 90𝑥

𝟗𝒙

(𝟗)(𝟕 + 𝟓𝒃)

13.

3 2 −3𝑘 + 15𝑘 − 6𝑘

𝟐 (𝟑𝒌)(−𝒌 + 𝟓𝒌 − 𝟐)

𝟑 𝟑𝒏

𝟐 𝟐−𝒏

14.

3 2 50𝑝 + 50𝑝 − 20

𝟑 𝟐 (𝟏𝟎)(𝟓𝒑 + 𝟓𝒑 − 𝟐)

𝒓+𝟏

15.

𝟔

16.

−90𝑥 + 100𝑥 + 60

(𝟏𝟎)(−𝟗𝒙 + 𝟏𝟎𝒙 + 𝟔)

𝟐

17.

2 3𝑚 + 9𝑚 + 27

𝟐 (𝟑)(𝒎 + 𝟑𝒎 + 𝟗)

𝟐 (𝟒)(𝟑𝒓 + 𝒓 − 𝟑)

𝟐 (𝟖)(𝟖 + 𝟓𝒙 + 𝟗𝒙)

𝟑

4.

3

6𝑛 − 3𝑛

5

5.

5

−6𝑟 − 6𝑟

4

−𝟔𝒓

6.

12 6 63𝑥 − 35𝑥

𝟔

7.

2 3 14𝑎 + 21𝑎 + 21𝑎

8.

3 2 10𝑛 − 9𝑛 + 𝑛

𝒏

𝟐 𝟏𝟎𝒏 − 𝟗𝒏 + 𝟏

18.

2 12𝑟 + 4𝑟 − 12

9.

2 −28𝑣 − 8𝑣 − 36

𝟐 (−𝟒)(𝟕𝒗 + 𝟐𝒗 + 𝟗)

19.

2 64 + 40𝑥 + 72𝑥

10.

7 6 5 −8𝑥 + 24𝑥 + 12𝑥

𝟒

𝟕𝒙

𝟗𝒙 − 𝟓

𝟕𝒂 𝟐 + 𝟑𝒂 + 𝟑𝒂

𝟓

Factors

2 3 20 − 35𝑛 − 20𝑛

𝟒𝒑

9𝑥 + 36

Given

𝟐

𝟒𝒙 (−𝟐𝒙 + 𝟔𝒙 + 𝟑)

20.

𝟐

𝟑

𝟓 𝟒 − 𝟕𝒏 − 𝟒𝒏

𝟐

𝟒

𝒙 − 𝟕𝒙 − 𝟏𝟎

3

𝟑 (𝟒)(𝟖𝒏 + 𝟕𝒏 − 𝟓)

5

𝟓

32𝑛 + 28𝑛 − 20

2

−18𝑛 + 15𝑛 − 15

𝟐 (𝟑)(−𝟔𝒏 + 𝟓𝒏 − 𝟓)

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LESSON 001 GRADE 08 MATHEMATICS: ALGEBRA

Factoring Polynomials with Common Monomial Factor

How can I factor polynomials with common monomial factor?

85.87 cm

Written, Prepared, and Edited by:

John michael s. cariazo, lpt

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