Factoring by Greatest Common Monomial Factor Subject Teacher: Saturnino P. Odon Jr. Activity: Pieces of My Life Directions: Find the possible factors of the given number or expression below. Choose you answers from the box and write it your answer sheet. 2 x z a y 5 10 6 4 b 3 Number/Expression 1.8 2.2x 3.5ab 4.12z 5.20xy Factors 1. 2. 3. 4. 5. 2, 4 2, x 5, a, b 2, 3, 4, 6, z 2, 10, 5, 4, x, y Questions: The area of a rectangle is the product of the length and the width, or 𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝐿 ⋅ 𝑊 1. What is the area of the rectangle? 2. Is the area of the rectangle a polynomial? 3. What is the relationship between the area of the rectangle and its sides? 4. What can you say about the width of the rectangle comparing it to the area? 5. What do you call the process of rewriting the polynomial as a product of polynomial factors? FACTORING OUT GCMF Factor Completely – to express as the product of prime factors Factor the following completely: 1) 5x2 2) 14x2y3 1) 5x2 5 Example: Factor completely 24 24 x x 14x2y3 2) 6 4 2 3 2 2 Therefore the factors of 24 are 2 2 2 3 5xx x2 14 2 7 x x2 y3 y x y y 27xxyyy GCF of 12 and 18 GCMF : Greatest Common Monomial Factor – The greatest monomial that is a factor (will divide EVENLY into) of all the given monomials. 12 = 1, 2, 3, 4, 6, 12 18 = 1, 2, 3, 4, 6, 9, 18 GCF = 6 GCF of 12x4y3 and 18xy5 12x4y3 = 1, 2, 3, 4, 6, 12, x, x, x, x, y, y, y 18xy5 = 1, 2, 3, 4, 6, 9, 18, x, y, y, y, y, y GCF = 6xy3 To find the GCMF of two or more Monomials •First find the GCF of the coefficients •Find the largest power of each variable that is COMMON to all the monomials •The GCMF = product of GCF of coefficients and common variable factors Example: Find the GCF of each pair of monomials. a. 4x3 and 8x2 b. 15y6 and 9z Solutions: a. 4x3 and 8x2 Step 1:Factor each monomial. 4x3 = 2 ⦁ 2 ⦁ x ⦁ x ⦁ x 8x2 = 2 ⦁ 2 ⦁ 2 ⦁ x ⦁ x Step 2. Identify the common factors. 4x3 = 2 ⦁ 2 ⦁ x ⦁ x ⦁ x 8x2 = 2 ⦁ 2 ⦁ 2 ⦁ x ⦁ x Step 3. Find the product of the common factors. 2 ⦁ 2 ⦁ x ⦁ x = 4x2 Therefore the GCMF of 4x3 and 8x2 is 4x2 b. 15y6 and 9z Activity 1: 1. Find GCMF of 12x2 and 18x 2. Find GCMF of 21x2 and 35x5 3. Find GCMF of 24x2y3 and 36x3y Solutions: 1. Find GCMF of 12x2 and 18x GCF of coefficients = 6 Common variable(s) : only have one x in common GCMF = 6x 2. Find GCMF of 21x2 and 35x5 GCF of Coefficients = 7 Common variable factors : two x’s GCMF = 7x2 3. Find GCMF of 24x2y3 and 36x3y GCF of coefficients = 12 Common variable factors : two x’s and one y GCMF = 12x2y Notice that in the examples above, prime factorization is used to find the GCMF of the given pair of monomials. The next examples illustrate how the GCMF is used to factor polynomials. NOW, we are going to use GCMF’s to Factor Quadratic Expressions. Factoring Out the GCMF is the inverse (un-doing) of the Distributive Property To factor – undoing distributive property 1) Perform Distributive Property 6(2 + 3) = 12 + 18 Factor : 12 + 18 = 6(2 + 3) 2) Use Distributive Property to simplify 3(x + 7) = 3x + 21 Factor: 3x + 21 = 3(x + 7) 3) Factor: 12x2y – 14xy3 = 2xy(6x – 7y2) Ex.Distribute 3x(x + 5) Means to multiply the 3x through the (x + 5) 3x(x) + 3x(5) 3x2 + 15x Ex. Factor 3x2 + 15x Means to Divide the GCMF out of the polynomial (divide each term by GCMF) GCMF = 3x Recall how to divide by monomial Divide (3x2 + 15x) by GCMF (3x) 2 3 x 15 x 3 x x 3 5 x 3 x 15 x x5 3x 3x 3 x 3 x 3x 2 Factored form is 3x(x + 5) To factor a polynomial by factoring out the GCMF: Example: Factor 15x2 – 9 1) Find the GCMF Step 1) GCMF = 3 2) Divide the polynomial (each term of the polynomial) by the GCMF Step 2) Divide 15x2 – 9 by the GCMF 3) Write the polynomial as the product of the GCMF and the result from step #2 Solution: 15 x 9 15 x 9 5x 2 3 3 3 3 2 2 Step 3) Write as a product of GCMF and result of step 2 3(5x2 – 3) Another Example!!! 1. Write 6𝑥+ 3x2 in factored form. Step 1 Determine the number of terms. In the given expression, we have 2 terms: 6𝑥 and 3x2. Determine the GCF of the numerical coefficients. Step 2 Coefficient 3 Factors 1, 3 6 1, 2, 3 Step 3 Common Factors GCF 1 and 3 3 Determine the GCF of the variables. The GCF of the variables is the one with the least exponent. GCF (x, x2) = x Another Example!!! 1. Write 6𝑥+ 3x2 in factored form. Step 4 Find the product of GCF of the numerical coefficient and the variables. (3)(x) = 3x Hence, 3𝑥 is the GCMF of 6𝑥 and 3x2. Step 5 Find the other factor, by dividing each term of the polynomial 6𝑥+ 3x2 by the GCMF 3𝑥. Step 6 Write the complete factored form 6𝑥 + 3x2 = 3x(2 + x) Activity 2: Factor the following expressions. 1. 28a3-12a2 2. 15a – 25b + 20 3. 16x5 – 14x3 + 26x2 Factor 1) 28a3-12a2 2) GCMF = 4a2 Factored Form 4a2(7a – 3) 15a – 25b + 20 GCMF = 5 Factored Form 5(3a-5b+4) 3) 16x5 – 14x3 + 26x2 GCMF = 2x2 Factored Form 2x2(8x3 – 7x + 13)