Grade 08 Mathematics: Factoring Polynomials with Difference of Two Squares

LESSON 002 GRADE 08 MATHEMATICS: ALGEBRA

Factoring Polynomials with Difference of Two Squares 85.87 cm

How can I factor polynomials with difference of two squares?

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

A.Unlocking Key Concepts

DIFFERENCE OF TWO SQUARES A.DIFFERENCE

The operation involved between terms should be subtraction.

B. TWO

There are only TWO terms in a polynomial involved.

85.87 cm

C. SQUARES

𝑛

The terms can be expressed in a form of 𝑥 , where n is even

GENERAL DEFINITION

𝑛

𝑎𝑥 = 𝑛

𝑎𝑥 =

𝑛 𝑎𝑥

𝑛 2 𝑎𝑥 2

TRY THESE!

a. 16𝑥

2

b. 25𝑥

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4

2

c. 11𝑥

8

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

A.Unlocking Key Concepts ANSWERS:

a. 16𝑥

2

b. 25𝑥 c. 11𝑥

4

8

TRY THESE!

REMEMBER WHAT IS SQUARE ROOT?

= 16 = 4 1 2 = 𝑥 =𝑥 =𝑥

= 25 = 5 2 4 = 𝑥 =𝑥 = 11 = 11 4 8 = 𝑥 =𝑥

4𝑥 5𝑥

2

The square root of a number is defined as the value, which gives the number when it is multiplied by itself.

While the square root of a variable is defined as a variable with half of its exponent.

2 2

EXAMPLES:

11𝑥

4

25 = 5

10𝑥 4 = 10𝑥 2

𝑥4 = 𝑥2

8𝑥 6 = 2 2𝑥 3

16𝑥 2 = 4𝑥

90𝑥 10 = 3 10𝑥 5

7= 7

160𝑥 20 = 4 10𝑥 10

2

THINGS TO REMEMBER

25 = 5 15 = 3.87 …

15 = 15

50 = 25 ∙ 2 = 25 ∙ 2 = 5 2 jmathsph@gmail.com https://www.facebook.com/jmathsinc/ https://issuu.com/mathmadeeasyph

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

A.Unlocking Key Concepts THINGS TO REMEMBER

3 6 𝑥 =𝑥

2 4 2 𝑥 𝑦 =𝑥 𝑦

−5 −10 𝑥 =𝑥

5 10 𝑥 =𝑥

5 𝑥 =𝑥

5 2

−7 𝑥 =𝑥

5 𝑥 =

5 𝑥

−7 𝑥 =

−7 2

−7 𝑥

B.Introducing New Concepts NEW CONCEPTS

Identifying Polynomials with Difference of Two Squares

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

B.Introducing New Concepts

TRY THESE!

EXAMPLES GIVEN

IS IT A BINOMIAL?

CAN BOTH TERMS BE CONVERTED INTO A SQUARE?

IS THE FIRST TERM POSITIVE AND THE OTHER TERM NEGATIVE, OR VICE VERSA?

CAN YOU USE TECHNIQUE NUMBER 2?

2

49𝑏 − 25 2 121𝑛 − 25 2 169𝑛 − 9 2 49𝑘 + 81 2 −9𝑥 + 25 2 49𝑛 + 9 3 𝑢 +4 2 3 𝑥 +𝑦 2 100𝑟 − 1 4 6 4𝑥 − 𝑦

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

B.Introducing New Concepts TRY THESE!

ANSWER KEY GIVEN

IS IT A BINOMIAL?

CAN BOTH TERMS BE CONVERTED INTO A SQUARE?

2

YES YES YES YES YES YES YES YES YES YES

YES YES YES YES YES YES NO NO YES YES

49𝑏 − 25 2 121𝑛 − 25 2 169𝑛 − 9 2 49𝑘 + 81 2 −9𝑥 + 25 2 49𝑛 + 9 3 𝑢 +4 2 3 𝑥 +𝑦 2 100𝑟 − 1 4 6 4𝑥 − 𝑦

IS THE FIRST TERM POSITIVE AND THE OTHER TERM NEGATIVE, OR VICE VERSA?

CAN YOU USE TECHNIQUE NUMBER 2?

YES YES YES NO YES NO NO NO YES YES

YES YES YES NO YES NO NO NO YES YES

Factoring Polynomials with Difference of Two Squares

STEP 1

STEP 2

STEP 3

STEP 4

Arrange the variables

Prepare your Answer

Take the Square Roots

Finalized Your Answer

Arrange the binomial, where the first term should be POSITIVE, and the second term should be NEGATIVE. 2 2 (𝑎 − 𝑏 )

The factors of this type of polynomial will always be in this form:

Take the square roots of both the terms regardless of the symbol (positive or negative)

(___ + ___) (___ - ___)

2

𝑎2 = 𝑎

Place the square root of the positive term on the first blanks in Step 2, and the square root of the negative term on the remaining blanks.

𝑏2

𝑏2 = 𝑏

𝑎 2

𝑎2 =

𝑎

𝑏2 = 𝑏

2

(𝑎 − 𝑏 )

POSITIVE

NEGATIVE

ADDITION SUBTRACTION SYMBOL SYMBOL

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𝑎 - ___) 𝑎 + ___) 𝑏 (___ 𝑏 (___

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

B.Introducing New Concepts TRY THESE!

EXAMPLE NO. 1 2

49𝑏 − 25 2 = 49𝑏 − 25 =( + )( − 2 = 49𝑏 = 𝑏 = 25 = 5 = (𝑏 + 5)(𝑏 − 5)

)

EXAMPLE NO. 2 2

−1 + 100𝑟 2 = 100𝑟 − 1 =( + )( − ) 2 = 100𝑟 = 10𝑟 = 1=1 = (10𝑟 + 1)(10𝑟 − 1) EXAMPLE NO. 3 2 10

2 4

-16𝑦 𝑥 + 4𝑥 𝑦 2 4 2 10 = 4𝑥 𝑦 − 16𝑦 𝑥 =( + )( − ) 2 2 4 = 4𝑥 𝑦 = 2𝑥𝑦 5 2 10 = 16𝑦 𝑥 = 4𝑦𝑥 2 5 2 5 = 2𝑥𝑦 − 4𝑦𝑥 2𝑥𝑦 − 4𝑦𝑥 jmathsph@gmail.com https://www.facebook.com/jmathsinc/ https://issuu.com/mathmadeeasyph

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

B.Introducing New Concepts TRY THESE!

EXAMPLES 2

2

2𝑥 − 18

196𝑛 − 144

2

2

= 2(𝑥 − 9)

= 4(49𝑛 − 36)

= 2(𝑥 − 3)(𝑥 + 3)

= 4(7𝑛 − 6)(7𝑛 + 6)

4

4

4

2𝑥 𝑟 − 72𝑦 𝑟 4

125𝑚 − 20𝑛 4

4

= 5 25𝑚 − 4𝑛

= 2𝑟(𝑥 − 36𝑦 ) 2

2

2

2

= 2𝑟(𝑥 − 6𝑦 )(𝑥 + 6𝑦 )

Factoring Polynomials 4 5 216𝑥 𝑎𝑦 − 6𝑦 𝑎 with Difference of Two Squares 4

= 6𝑎𝑦 6𝑥 − 𝑦

2

= 5 5𝑚 − 2𝑛

4

2

2

2

2

2

5𝑚 + 2𝑛

4

4

4𝑥 𝑚 − 36𝑦 𝑚 4

4

4

= 𝑚(4𝑥 − 36𝑦 )

= 6𝑎𝑦(36𝑥 − 𝑦 ) 2

2

4

2

6𝑥 + 𝑦

2

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2

2

= 𝑚(2𝑥 − 6𝑦 )(2𝑥 + 6𝑦 )

GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

C. More Examples TRY THESE!

ACTIVITY Given

Factors

Given

Factors

1.

2 9𝑥 − 1

11.

2 2 𝑎 − 25𝑏

2.

2 4𝑛 − 49

12.

2 2 6𝑎 − 96𝑏

3.

2 36𝑘 − 1

13.

2 2 𝑥 − 9𝑦

4.

2 𝑝 − 36

14.

2 2 54𝑣 − 6𝑢

5.

2

180𝑚 − 5

15.

36𝑎 − 25𝑏

6.

2 294𝑟 − 150

16.

4 4 4𝑥 − 144𝑦

7.

2 150𝑘 − 216

17.

4 4 7𝑥 − 28𝑦

8.

2 20𝑎 − 45

18.

4 4 7𝑥 − 343𝑦

9.

2 3𝑛 − 75

19.

6 6 16𝑚 − 𝑛

20.

6

10.

3

24𝑥 − 54𝑥

4

64𝑥 − 𝑦

4

6

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GRADE 08 MATHEMATICS: ALGEBRA Factoring Polynomials with Difference of Two Squares

C. More Examples TRY THESE!

ACTIVITY Given 1.

2 9𝑥 − 1

2.

2 4𝑛 − 49

3.

2 36𝑘 − 1

4.

2 𝑝 − 36

5.

2

180𝑚 − 5

6.

2 294𝑟 − 150

7.

2 150𝑘 − 216

8.

2 20𝑎 − 45

9.

2 3𝑛 − 75

10.

3

24𝑥 − 54𝑥

Factors

Given

Factors (𝒂 + 𝟓𝒃)(𝒂 − 𝟓𝒃)

(𝟑𝒙 + 𝟏)(𝟑𝒙 − 𝟏)

11.

2 2 𝑎 − 25𝑏

(𝟐𝒏 + 𝟕)(𝟐𝒏 − 𝟕)

12.

2 2 6𝑎 − 96𝑏

𝟔(𝒂 + 𝟒𝒃)(𝒂 − 𝟒𝒃)

(𝟔𝒌 + 𝟏)(𝟔𝒌 − 𝟏)

13.

2 2 𝑥 − 9𝑦

(𝒙 + 𝟑𝒚)(𝒙 − 𝟑𝒚)

(𝒑 + 𝟔)(𝒑 − 𝟔)

14.

2 2 54𝑣 − 6𝑢

𝟔(𝟑𝒗 + 𝒖)(𝟑𝒗 − 𝒖)

𝟓(𝟔𝒎 + 𝟏)(𝟔𝒎 − 𝟏)

15.

36𝑎 − 25𝑏

4

𝟐 𝟐 𝟐 𝟐 (𝟔𝒂 + 𝟓𝒃 )(𝟔𝒂 − 𝟓𝒃 )

𝟔(𝟕𝒓 + 𝟓)(𝟕𝒓 − 𝟓)

16.

4 4 4𝑥 − 144𝑦

𝟐 𝟐 𝟐 𝟐 𝟒(𝒙 + 𝟔𝒚 )(𝒙 − 𝟔𝒚 )

𝟔(𝟓𝒌 + 𝟔)(𝟓𝒌 − 𝟔)

17.

4 4 7𝑥 − 28𝑦

𝟐 𝟐 𝟐 𝟐 𝟕(𝒙 + 𝟐𝒚 )(𝒙 − 𝟐𝒚 )

𝟓(𝟐𝒂 + 𝟑)(𝟐𝒂 − 𝟑)

18.

4 4 7𝑥 − 343𝑦

𝟐 𝟐 𝟐 𝟐 𝟕(𝒙 + 𝟕𝒚 )(𝒙 − 𝟕𝒚 )

𝟑(𝒏 + 𝟓)(𝒏 − 𝟓)

19.

6 6 16𝑚 − 𝑛

𝟑 𝟑 𝟑 𝟑 (𝟒𝒎 + 𝒏 )(𝟒𝒎 − 𝒏 )

𝟔𝒙(𝟐𝒙 + 𝟑)(𝟐𝒙 − 𝟑)

20.

6

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4

64𝑥 − 𝑦

6

𝟑

𝟑

𝟖𝒙 + 𝒚

𝟑

𝟑

𝟖𝒙 − 𝒚

LESSON 002 GRADE 08 MATHEMATICS: ALGEBRA

Factoring Polynomials with Difference of Two Squares

How can I factor polynomials with difference of two squares?

85.87 cm

Written, Prepared, and Edited by:

John Michael s. cariazo, lpt

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