Factoring Polynomials with Common Monomial Factor.pptx

MATH 8 - FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR

1. FACTORING of polynomials Module 1

2. FACTORING techniques: Lesson 1:COMMON MONOMIAL factoring Lesson 2: DIFFERENCE OF TWO SQUARES Lesson 3: SUM AND DIFFERENCE OF TWO CUBES

3. Lesson 4: PERFECT SQUARE TRINOMIAL Lesson 5: GENERAL QUADRATIC TRINOMIAL, where a = 1 Lesson 6: GENERAL QUADRATIC TRINOMIAL, where a >1 Lesson 7: FACTORING BY GROUPING

4. melc (Most Essential learning competency) 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). LC Code: M8AL-Ia-b-1

5. a. Factoring polynomial with Greatest common monomiaL Objectives: 1.find the greatest common monomial factor (GCMF) of polynomials 2.factor polynomials with greatest common monomial factor (GCMF) completely.

6. DEFINITION OF TERMS Factoring is the process of finding factors of a given product. It is the reverse of multiplication. 6 = 1 ● 6 6 = Product 2, 3 = Factors 6 = 2 ● 3 1, 6 = Factors 6 = Product

7. There are three techniques you can use for MULTIPLYING POLYNOMIALS. Distributive Property 2. FOIL Method 3. Box Method

8. Recall::: 1. Distributive property Example 1: 4(3a + 2) 4(3a + 2) = 12a + 8 Example 2: -2x 2 (2x – 1) -2x 2 (2x – 1) = -4x 3 + 2x 2 Example 3: 3x 2 y(4x 3 y – 5) 3x 2 y(4x 3 y – 5)= 12x 5 y 2 – 15x 2 y

9. VERTICAL METHOD in MULTIPLYING BINOMIAL                     24 6x + 24 6x + 24 4x 6x + 24 +4x   6x + 24 + 4x   + 10x + 24  

10. F O I L Made of four multiplication steps FIRST OUTER INNER LAST TERMS This method can only be used when we are multiplying a binomial by a binomial.

11. RECALL on SQUARE OF BINOMIAL Example 1:   2. USING FOIL METHOD   =   FIRST: a ▪ a =   OUTER : a ▪ 4 = 4a F = FIRST O = OUTER I = INNER L = LAST INNER : 4 ▪ a = 4a LAST : 4 ▪ 4 = 16 8a

12. EXAMPLE Given: Solution First First Last Outer Inner Inner Outer Last

13. 3. box method By using this method we can multiply any two polynomials. In this method we need to draw a box which contains some rows and columns. the number of rows is up to the number of terms of the first polynomial and the number of columns is up to the number of terms of the second polynomial.

14. EXAMPLE Given: The 1 st  and 2 nd  polynomial is containing two terms, so the number of rows and number of columns in the box must be 2. Step 1 Step 2 Combine Like Terms 12  

15. EXAMPLE Given: The 1 st   polynomial is containing two terms and 2 nd polynomial is containing three terms, so the number of rows in the box is 2 and the number of columns in the box must be 3. Step 1 Step 2 Combine Like Terms  

16. factoring is t he process of finding the factors of a mathematical expression or is the reverse process of multiplication. ab + ac = a( b+c )

17. Greatest common monomial FACTOR Formula : ax + bx + cx = x (a + b+ c) The Greatest Common Factor (GCF) is the largest value of a number, a variable, or a combination of numbers and variables which is common in each term of a given polynomial.

18. Greatest common FACTOR Find the GCF of the following numbers. Write the prime factorization of each numbers. a. 16 and 28 = 16 = 4 ▪ 4 = 2 ▪ 2 ▪ 2 ▪ 2 28 = 7 ▪ 4 = 7 ▪ 2 ▪ 2 Greatest Common Factor = 2 ▪ 2 = 4

19. b.   = a ▪ a ▪ b ▪ c ▪c   = a ▪ a ▪ b ▪ b ▪ c▪ c ▪ c   Greatest Common Factor = a ▪ a ▪ b ▪ c ▪ c =  

20. C.   = 3 ▪ 2 ▪ 2 ▪ x ▪ x ▪ y ▪ y ▪ y ▪ z ▪ z   36 = 3 ▪ 2 ▪ 2 ▪ 3 ▪ x ▪ x ▪ y ▪ z   Greatest Common Factor 3 ▪ 2 ▪ 2 ▪ x ▪ x ▪ y ▪ z = 12 yz   NOTE : In choosing the Greatest Common Factor, use the following rules. Solve for the GCF for the constants. Get the common variable with the lowest exponent.

21. Steps in factoring polynomials with Greatest Common Monomial Factor (GCMF): Find the GCF. Divide each term in the polynomial by its GCMF. Combine the answers in Steps 1 and 2 as a product . STEPS :

22. EXAMPLE 1 a. Find the GCF of 6x and 4 x² GCMF = 2 x b. Divide each term in the polynomial by its GCMF. 6 x + 4x² = 3 + 2x 2 x 2 x c. Combine the answers in Steps 1 and 2 as a product. Factor 6x + 4 x² 2 x ( 3 + 2x) To check, apply the distributive property. 2 x ( 3 + 2x) = 6x + 4 x² Therefore, the factors are 2 x ( 3 + 2x).

23. EXAMPLE 2 a. Find the GCF of 12 x²y 3 z² and 36x²yz GCMF = 12 x²yz b. Divide each term in the polynomial by its GCMF. 12 x²y 3 z² + 36x²yz = y 2 z + 3 12 x²yz 12 x²yz c. Combine the answers in Steps 1 and 2 as a product. Factor 12 x²y 3 z² + 36x²yz 12 x²yz ( y 2 z + 3) To check, apply the distributive property. 12 x²yz ( y 2 z + 3) = 12 x²y 3 z² + 36x²yz Therefore, the factors are 12 x²yz ( y 2 z + 3)

24. EXAMPLE 3 a. Find the GCF of 3a and 7b GCMF = 1 b. Divide each term in the polynomial by its GCMF. 3a + 7b =3a + 7b 1 1 c. Combine the answers in Steps 1 and 2 as a product. Factor 3a + 7b Prime polynomials are polynomials cannot be factored. Other examples of prime polynomials are 5+3𝑏, 2𝑥−7𝑦 and 𝑎+2𝑏+3𝑐. This polynomial cannot be factored by removing the common factor since the GCF in each term is 1. Polynomial of this type is called Prime Polynomial.

25. exAMPLE 4 Factor x²yz + xy²z + xyz² Solution: GCF = xyz x²yz + xy²z + xyz² = x + y+ z xyz x²yz +xy²z +xyz² = xyz (x + y + z)

26. EXAMPLE 5 Factor 14m²n² - 4mn³ Solution: GCF = 2mn² 14m²n² - 4mn³ = 2mn² (7m – 2n) Note: remember to write the sign for addition (+) or subtraction (-) when dividing. 14m²n² – 4mn³ = 7m – 2n 2mn² 2mn²