# Q1 week 1 (common monomial,sum &amp; diffrence of two cubes,difference of two squares)

Walden MacabuhayFaculty / ICT Teacher em The College of Maasin

It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.

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### Q1 week 1 (common monomial,sum &amp; diffrence of two cubes,difference of two squares)

• 1. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Quarter One: Patterns and Algebra Topic: Factor of Polynomials Let us recall the distributive property which state that if a, b, and c are real numbers, then 𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐) If we keep the distributive property in mind, it will not be difficult to factor a polynomial having a common monomial factor like 2𝑥3 + 𝑥2 − 7𝑥. We can easily observe that the three terms of 2𝑥3 + 𝑥2 − 7𝑥 have the common factor 𝑥. That is, 2𝑥3 + 𝑥2 − 7𝑥 = (2𝑥2)(𝑥) + (𝑥)(𝑥) − 7(𝑥) Thus, according to the distributive property, we may write this as 2𝑥3 + 𝑥2 − 7𝑥 = (2𝑥2)(𝑥) + (𝑥)(𝑥) − 7(𝑥) 2𝑥3 + 𝑥2 − 7𝑥 = 𝑥(2𝑥2 + 𝑥 − 7) Let’s Try! a. 10𝑥3 + 9𝑥2 + 4𝑥 b. 3𝑥6 + 9𝑥4 + 12𝑥2 A polynomial whose terms have a common monomial factor may be factored by identifying this common factor and applying the distributive property of multiplication over addition. 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). MELCs & Codes: M8AL-Ia-b-1 Common Monomial Factor Objective: Factor completely polynomials with common monomial factors Introduction
• 2. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Otherwise, Greatest Common Factor (GCF) may apply to ensure that the polynomial factor is irreducible or a prime polynomial. Greatest Common Factor (GCF) The greatest common factor is the largest integer, monomial, or multinomial that a set of numbers or polynomial have in common. For instance, Find the GCF of 12𝑥3 𝑦2 , 8𝑥𝑦2 , 𝑎𝑛𝑑 4𝑥2 𝑦2 . Solution: Express each as a product of prime factors. 12𝑥3 𝑦2 = 2 • 2 • 3 • x • x • x • y • y 8𝑥𝑦2 = 2 • 2 • 3 • x • y • y 4𝑥2 𝑦2 = 2 • 2 • x • x • y • y 2 2 x y y The GCF of these three monomials is 2 • 2 • x • y • y = 4𝑥𝑦2 . Notice that the degree of the GCF is equal to or less than the degree of the expression with the lowest degree. Let’s Try! a. Find the GCF of 24𝑎2 𝑏3 𝑐3 , 30𝑎3 𝑏𝑐4 , 𝑎𝑛𝑑 48𝑎𝑏2 𝑐2 . Factoring is the reverse process of multiplication. When a number or a polynomial is factored, it is written as a product of two or more factors. A polynomial is said to be factored into prime factors if it expresses as the product of two or more irreducible or prime polynomials of the same type. A polynomial is factored completely if each of its factors can no longer be expressed as a product of two other polynomials of lower degree and that the coefficients have no common factors without introducing a fraction, 1 or -1. If each term of a polynomials is divisible by the same monomial, this monomial is referred to as a Common Monomial Factor. Common Monomial Factoring 1. Find the greatest common factors (GCF) of the terms in the polynomials. This is the first factor. 2. Divide each term by the GCF to get the other factor.
• 3. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 v Example 1. Factors each expression a. 7𝑥2 − 7𝑦 b. 8𝑥3 − 16𝑥4 + 48𝑥7 c. 2(𝑎 + 𝑏) − 𝑥(𝑎 + 𝑏) Solution: a. The GCF of 7𝑥2 and −7𝑦is 7. 7𝑥2 − 7𝑦 = 7 ( 7𝑥2 7 + −7𝑦 7 ) = 7(𝑥2 − 𝑦) b. The GCF of 8𝑥3 , −16𝑥4 and 48𝑥7 is 8𝑥3 . 8𝑥3 = 2 • 2 • 2 • x • x • x 16𝑥4 = 2 • 2 • 2 • x • x • x • x • 2 48𝑥7 = 2 • 2 • 2 • x • x • x • x • 2 • 3 • x • x • x 2 • 2 • 2 • x • x • x GCF is 2 • 2 • 2 • x • x • x = 8𝑥3 Factor by Common Monomial: 8𝑥3 − 16𝑥4 + 48𝑥7 = 8𝑥3 ( 8𝑥3 8𝑥3 − 16𝑥4 8𝑥3 + 48𝑥7 8𝑥3 ) = 8𝑥3(1 − 2𝑥 + 6𝑥4) c.    baxba 2   xba  2 Give the GCF of the given monomials. 1. 85,70,45 2. 6635 , yxyx 3. 524435 26,110,98 yxyxyx 4. yzxyxyz 55,55,15 2  5. 453435 48,48,32 yxyxyx  Activities
• 4. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Factor the following polynomials. 1. 𝑎2 𝑏𝑐 + 𝑎𝑏2 𝑐 + 𝑎𝑏𝑐2 2. 4𝑚2 𝑛2 − 4𝑚𝑛3 3. 25𝑎 + 25𝑏 4. 3𝑥2 + 9𝑥𝑦 5. 2𝑥2 𝑦 + 12𝑥𝑦 A. Reflect on your participation in doing all the activities in this lesson and complete the following statements: • I learned that I... • I was surprised that I... • I noticed that I... • I discovered that I... • I was pleased that I... Exercise JOURNAL WRITING: “Common Monomial Factor” Description: This journal will enable you to reflect about the topic and activities you underwent. Instruction: Reflect on the activities you have done in this lesson by completing the following statements. Write your answers on the space provided for. References: Orines, Fernando B. Mathematics 8. Next Century second Edition Bureau of Secondary Education. Distance Learning Module Mathematics 2 Escaner, Jose Maria L. IV PhD. et.al (2013) K to 12 spiral math 8
• 5. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Name: ____________________ Grade & Section: ______________ Instruction: Reflect on your participation in doing all the activities in this lesson and complete the following statements. Write your answers on the space provided for.  I learned that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was surprised that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I noticed that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I discovered that I… _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was pleased that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ “JOURNAL WRITING” Common Monomial Factor
• 6. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Quarter One: Patterns and Algebra Topic: Factor of Polynomials Let’s recall the special pattern 𝑎2 − 𝑏2 , which is the result when the sum of two terms is multiplied by the difference of the same two terms. In other words, when the two binomials have the form (𝑎 + 𝑏) and (𝑎 − 𝑏), you can easily get the product as (𝑎2 − 𝑏2) which is the difference of 2 perfect squares. For example, (𝑥 + 5)(𝑥 − 5) = 𝑥2 − 25. Therefore, whenever you encounter a binomial that has the form 𝑎2 − 𝑏2 , you can do the reverse process where in the given terms are both perfect squares. Say, 𝑥2 − 25 = (𝑥)2 − (5)2 = (𝑥 + 5)(𝑥 − 5) Factoring the Difference of two squares is a special type of factoring, a problem that is often used in mathematics 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). MELCs & Codes: M8AL-Ia-b-1 Factors of Difference of Two Squares Objective: Factor completely polynomials with difference of two squares. Introduction Factors of Difference of Two Squares 1. Get the principal square root of each of the two squares. 2. Using these square roots, form two factors: a sum and a difference.
• 7. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 This pattern can be generalized as follows: A binomial that is the difference between two squares, 𝑎2 − 𝑏2 , for any real numbers, a and b, can be factored as the product of the sum (𝑎 + 𝑏) and the difference (𝑎 − 𝑏) of the terms that are being squared: 𝑎2 − 𝑏2 = (𝑎 + 𝑏) (𝑎 − 𝑏) Examples 1. Factor the following polynomials a. 𝑥2 − 36 b. 4𝑎2 − 9𝑏2 Solution: a. 𝑥2 − 36 𝑥2 − 36 = (𝑥)2 − (6)2 , therefore, a=x and b=6 𝑎2 − 𝑏2 = (𝑎 + 𝑏) (𝑎 − 𝑏) use difference of two squares pattern 𝑥2 − 62 = (𝑥 + 6) (𝑥 − 6), by substitution 𝑥2 − 36 = (𝑥 + 6) (𝑥 − 6) b. 4𝑎2 − 9𝑏2 4𝑎2 − 9𝑏2 = (2𝑎)2 − (3𝑏)2 , therefore, a=2a and b=3b 𝑎2 − 𝑏2 = (𝑎 + 𝑏) (𝑎 − 𝑏) use difference of two squares pattern (2𝑎)2 − (3𝑏)2 = (2𝑎 + 3𝑏) (2𝑎 − 3𝑏), by substitution 4𝑎2 − 9𝑏2 = (2𝑎 + 3𝑏) (2𝑎 − 3𝑏) Give the square root of each. 1. 644 c 2. 8116 2 b 3. 462 10036 mkj  4. 24 4 ed Activities
• 8. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Factor the following polynomials 1. 9𝑎2 − 1 2. 81 − 625𝑧6 3. 121𝑔2 − 169ℎ4 4. ( 𝑥2 − 1)2 − 𝑥2 A. Reflect on your participation in doing all the activities in this lesson and complete the following statements: • I learned that I... • I was surprised that I... • I noticed that I... • I discovered that I... • I was pleased that I... Exercise JOURNAL WRITING: “Difference of Two Squares” Description: This journal will enable you to reflect about the topic and activities you underwent. Instruction: Reflect on the activities you have done in this lesson by completing the following statements. Write your answers on the space provided for. References: Orines, Fernando B. Mathematics 8. Next Century second Edition Bureau of Secondary Education. Distance Learning Module Mathematics 2 Escaner, Jose Maria L. IV PhD. et.al (2013) K to 12 spiral math 8
• 9. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Name: ____________________ Grade & Section: ______________ Instruction: Reflect on your participation in doing all the activities in this lesson and complete the following statements. Write your answers on the space provided for.  I learned that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was surprised that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I noticed that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I discovered that I… _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was pleased that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ “JOURNAL WRITING” Difference of Two Squares
• 10. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Sum or Difference of two cubes   2233 yxyxyxyx    2233 yxyxyxyx  Quarter One: Patterns and Algebra Topic: Factor of Polynomials Observe how the factors of 33 yx  are obtained by introducing arbitrary terms without affecting the given expression and by using grouping techniques. 3333 0 yxyx  Renaming 0 as sum of yx2 and yx2  3223 yyxyxx     3223 yyxyxx  Grouping the 1st two terms and the last two terms    222 yxyyxx  Bringing out the common monomial factors in each group     yxyxyyxx  2 Factoring the difference of two squares     yxyxyx  2 Factoring out  ,yx  a common binomial factor   22 yxyxyx  Follow the same process for 33 yx  to obtain   2233 yxyxyxyx  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). MELCs & Codes: M8AL-Ia-b-1 Sum and Difference of Two Cubes Objective: Factor completely polynomials with sum & difference of two cubes. Introduction
• 11. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Steps in factoring the sum or difference of two cubes 1. Get the cube root of each cubed terms 2. Taking the operation between the cubes, use the cube roots in step 1 to obtain a binomial factors. 3. Form the trinomial factor as follow: a. Square the first cube root. b. Multiply the two cubes roots. The sign of the product is opposite the sign between the cubes. c. Square the second cube root. Examples 1. 83 x Solution: Rename each terms as a sum of two cubes. Then, apply the formula for sum of cubes.    333 28  xx         22 222  xxx   422 2  xxx Example 2. 612 64yx  Solution: Note that the binomial is a difference of squares.    3234612 464 yxyx          22242424 444 yyxxyx    424824 1644 yyxxyx     424822 16422 yyxxyxyx  Factors the following expression. 1. 33 ax  3. 33 125 vu  5. 126 jh  2. 127 3 z 4. 333 rqp  Activities
• 12. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Factor the following expressions. 1. 6 8 27 y 3. 3 125216 d 5. 63 125.0008.0 dc  2. 366 8cba  4. 36 648 ts  A. Reflect on your participation in doing all the activities in this lesson and complete the following statements: • I learned that I... • I was surprised that I... • I noticed that I... • I discovered that I... • I was pleased that I... Exercise JOURNAL WRITING: “Sum or Difference of Two Cubes” Description: This journal will enable you to reflect about the topic and activities you underwent. Instruction: Reflect on the activities you have done in this lesson by completing the following statements. Write your answers on the space provided for. References: Orines, Fernando B. Mathematics 8. Next Century second Edition Bureau of Secondary Education. Distance Learning Module Mathematics 2 Escaner, Jose Maria L. IV PhD. et.al (2013) K to 12 spiral math 8
• 13. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Learning Insights Name: ____________________ Grade & Section: ______________ Instruction: Reflect on your participation in doing all the activities in this lesson and complete the following statements. Write your answers on the space provided for.  I learned that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was surprised that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I noticed that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I discovered that I… _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________  I was pleased that I... _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ “JOURNAL WRITING” Sum or Difference of Two Cubes
• 14. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Name: ____________________ Grade & Section: ______________ A. Give the GCF of the given monomials. 1. 77,49,35 2. 24927 , zyxzyx 3. 244337 72,42,18 yzxzyxzyx 4. 3273 36,18,9 yxyyx  5. 453434 168,84,24 qxyxyx B. Factor each expression. 1. 621459 x 2. 432332 544236 zyzyzy  3. 2275 4088104 vuuvvu  4. mkhhkgjhg 3723232  5. 344645 710535 bababa  “ASSESSMENT” Common Monomial Factors
• 15. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Name: ____________________ Grade & Section: ______________ Factor the following polynomials 1. 44 7527 wv  2. 82 49.0 gf 3. 23 3625 hjh  4. 1 25 2  a 5. 296,1256 2 y 6. 45 483 tut  7. 32 5424 mmk  8. 1282 16 x 9. 1002 n m 10. 50200 9850 yx  “ASSESSMENT” Difference of Two Squares
• 16. ROSE MARIEL F. MAITEM Mathematics Teacher, THE COLLEGE OF MAASIN Week 1: Mathematics 8 Name: ____________________ Grade & Section: ______________ A. Factor the following expressions. 1. 3 64 w 2. 16 t 3. 83 x 4. 6 125 a 5. 216963 fed 6. 1216 3 e 7. 33 64 1 8 gf  8. 123 8 xw  9. 2764 9 z 10. 36 001.0 fe  “ASSESSMENT” Sum or Difference of Two Cubes
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