Math 8: Factoring by Greatest Common Monomial Factor PDF

Lesson Factoring by Greatest 1 Common Monomial Factor One of the factoring techniques that you are going to learn in this module is factoring by greatest common monomial factor (GCMF). Concepts such as factors, factoring, and prime factorization have been...

Lesson Factoring by Greatest 1 Common Monomial Factor One of the factoring techniques that you are going to learn in this module is factoring by greatest common monomial factor (GCMF). Concepts such as factors, factoring, and prime factorization have been discussed and have been used in many instances in your previous math classes. Let us try to reactivate what you previously learned by answering the activity below. What’s In Recall that factor is a number or algebraic expression that divides another number or expressions evenly, that is with no remainder. Examples: 1. The factors of 4 are 1, 2, and 4 as these can divide 4 evenly. 2. The factors of 2𝑥 2 are 1, 2, 𝑥, 𝑥 2 , 2𝑥, 2𝑥 2 as these can divide 2𝑥 2 evenly. Activity: Pieces of My Life Find the possible factors of the given number or expression below. Choose you answers from the box and write it on your answer sheet. 2 x a 10 z y 6 b 5 4 3 Number/Expression Factors 1. 8 __________ 2. 2x __________ 3. 5ab __________ 4. 12z __________ 5. 20xy __________ 4 CO_Q1_Mathematics8_Module1A  What’s New Consider the rectangle below. 𝑳 = 𝒙 + 𝟑 𝑾 = 𝟐 The area of a rectangle is the product of the length and the width, or 𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝐿 ∙ 𝑊. Questions: 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? What is It Suppose we will make use of the area of the rectangle in the previous section which is 2𝑥 + 6. Now, working backward, we have to find the length and the width of the rectangle. Notice that 2𝑥 + 6 can be written as: 2 ∙𝑥+2 ∙3 Notice also that 2 is common to both terms. So, by rewriting it we have, 2𝑥 + 6 = 2 ∙ 𝑥 + 2 ∙ 3 = 2(𝑥 + 3) Recall that by distributive property, 2(𝑥 + 3) will go back to its original form 2𝑥 + 6. Hence, Note! When you factor, see to it the product 2(𝑥 + 3) = 2𝑥 + 6 of these factors is always the original expression or polynomial. 5 CO_Q1_Mathematics8_Module1A  This means that, 2(𝑥 + 3) is the completely factored form of 2𝑥 + 6. Based on the example above, you have noticed that the method of factoring used is finding a number or expression that is common to all the terms in the original expression, that is, 2 is a common factor to both 2𝑥 and 6. Since there is no other factor, other than 1, which is common to all terms in the given expression, 2 is called the greatest common monomial factor (GCMF). To further illustrate the concept of GCMF, try to explore the following examples: Example 1. Find the GCF of each pair of monomials. a. 4𝑥 3 𝑎𝑛𝑑 8𝑥 2 b. 15𝑦 6 𝑎𝑛𝑑 9𝑧 Solution: a. 4𝑥 3 𝑎𝑛𝑑 8𝑥 2 Step 1. Factor each monomial. 4𝑥 3 = 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 8𝑥 2 = 2 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 Step 2. Identify the common factors. 4𝑥 3 = 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 8𝑥 2 = 2 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 Step 3. Find the product of the common factors. 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 = 4𝑥 2 Hence, 4𝑥 2 is the GCMF of 4𝑥 3 𝑎𝑛𝑑 8𝑥 2. b. 15𝑦 6 𝑎𝑛𝑑 9𝑧 Step 1. Factor each monomial. 15𝑦 6 = 3 ∙ 5 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 9𝑧 = 3 ∙ 3 ∙ 𝑧 Step 2. Identify the common factors. 15𝑦 6 = 3 ∙ 5 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 9𝑧 = 3 ∙ 3 ∙ 𝑧 Step 3. Find the product of the common factors. Note that 3 is the only common factor. Hence, 3 is the GCMF of 15𝑦 6 𝑎𝑛𝑑 9𝑧 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. 6 CO_Q1_Mathematics8_Module1A Example 2. Write 6𝑥 + 3𝑥 2 in factored form. Step 1. Determine the number of terms. In the given expression, we have 2 terms: 6𝑥 and 3𝑥 2. Step 2. Determine the GCF of the numerical coefficients. coefficient factors common factors GCF 3 1, 3 1, 3 3 6 1, 2, 3 Step 2. Determine the GCF of the variables. The GCF of the variables is the one with the least exponent. 𝐺𝐶𝐹(𝑥, 𝑥 2 ) = 𝑥 Step 3. Find the product of GCF of the numerical coefficient and the variables. (3)(𝑥) = 3𝑥 Hence, 3𝑥 is the GCMF of 6𝑥 and 3𝑥 2. Step 4. Find the other factor, by dividing each term of the polynomial 6𝑥 + 3𝑥 2 by the GCMF 3𝑥. 6𝑥 3𝑥 2 → + Divide each term by the GCMF 3𝑥 3𝑥 3𝑥 ∙ 2 3𝑥 ∙ 𝑥 → + Rewrite each term as a product 3𝑥 3𝑥 → 𝟐+𝒙 Step 5. Write the complete factored form 6𝑥 + 3𝑥 2 = 𝟑𝒙 (𝟐 + 𝒙) Example 3: Write 12𝑥 3 𝑦 5 − 20𝑥 5 𝑦 2 𝑧 in complete factored form. Step 1. Determine the number of terms. There are two terms in the given expression 12𝑥 3 𝑦 5 − 20𝑥 5 𝑦 2 𝑧, 12𝑥 3 𝑦 5 𝑎𝑛𝑑 20𝑥 5 𝑦 2 𝑧. Step 2. Determine the GCF of the numerical coefficient. coefficient factors common factors GCF 12 1, 2, 3, 4, 6, 12 1, 2, 3, 4 4 20 1, 2, 4, 5, 10, 20 Step 2. Determine the GCF of the variables. The GCF of the variables is the one with the least exponent and is common to every term. 𝐺𝐶𝐹(𝑥 3 𝑦 5 , 𝑥 5 𝑦 2 𝑧) = 𝑥 3 𝑦 2 7 CO_Q1_Mathematics8_Module1A Step 3. Find the product of GCF of the numerical coefficient and the variables. 4 ∙ 𝑥3 𝑦2 = 4 𝑥3 𝑦2 This means that, 4 𝑥3 𝑦2 is the GCMF of the two terms 12𝑥 3 𝑦 5 𝑎𝑛𝑑 20𝑥 5 𝑦 2 𝑧. Step 4. Find the other factor, by dividing each term of the polynomial 12𝑥 3 𝑦 5 − 20𝑥 5 𝑦 2 𝑧 by the GCMF 4 𝑥 3 𝑦 2. 12𝑥 3 𝑦 5 20𝑥 5 𝑦 2 𝑧 → − 4 𝑥3 𝑦2 4 𝑥3 𝑦2 4𝑥 3 𝑦 2 ∙ 3𝑦 3 4𝑥 3 𝑦 2 ∙ 5𝑥 2 𝑧 → − 4 𝑥3 𝑦2 4 𝑥3 𝑦2 → 3𝑦 3 − 5𝑥 2 𝑧 Step 5. Write the complete factored form 12𝑥 3 𝑦 5 − 20𝑥 5 𝑦 2 𝑧 = 𝟒 𝒙𝟑 𝒚𝟐 (𝟑𝒚𝟑 − 𝟓𝒙𝟐 𝒛) Example 4: Write 12𝑥 3 − 18𝑥𝑦 + 24𝑥 in complete factored form. Step 1. Determine the number of terms. There are three terms in the expression 12𝑥 3 − 18𝑥𝑦 + 24𝑥: 12𝑥 3 , 18𝑥𝑦, 24𝑥 Step 2. Determine the GCF of the numerical coefficient. coefficient factors common factors GCF 12 1, 2, 3, 4, 6, 12 18 1, 2, 3, 6, 9, 18 1, 2, 3, 6 6 24 1, 2, 3, 4, 6, 8, 12, 24 Step 2. Determine the GCF of the variables. The GCF of the variables is the one with the least exponent and is common to every term. 𝐺𝐶𝐹( 𝑥 3 , 𝑥𝑦, 𝑥) = 𝑥 Step 3: Find the product of GCF of the numerical coefficient and the variables. (6)(𝑥) = 6𝑥 Hence, 6𝑥 is the GCMF of 12𝑥 3 , 18𝑥𝑦, 24𝑥. Step 4. Find the other factor, by dividing each term of the polynomial 12𝑥 3 − 18𝑥𝑦 + 24𝑥 by the GCMF 6𝑥. 12𝑥3 18𝑥𝑦 24𝑥 → − + 6𝑥 6𝑥 6𝑥 6𝑥 ∙ 2𝑥2 6𝑥 ∙ 3𝑦 6𝑥 ∙ 4 → − + 6𝑥 6𝑥 6𝑥 → 2𝑥 2 − 3𝑦 + 4 8 CO_Q1_Mathematics8_Module1A  Step 5: Write the complete factored form. 12𝑥 3 − 18𝑥𝑦 + 24𝑥 = 𝟔𝒙 (𝟐𝒙𝟐 − 𝟑𝒚 + 𝟒) Example 5. Write 28𝑥 3 𝑧 2 − 14𝑥 2 𝑦 3 + 36𝑦𝑧 4 in complete factored form. Step 1. Determine the number of terms. There are three terms in the expression 28𝑥 3 𝑧 2 − 14𝑥 2 𝑦 3 + 36𝑦𝑧 4 ∶ 28𝑥 3 𝑧 2 , 14𝑥 2 𝑦 3 , 𝑎𝑛𝑑 36𝑦𝑧 4. Step 2. Determine the GCF of the numerical coefficient. coefficient factors common factors GCF 28 1, 2, 4, 7, 14, 28 14 1, 2, 7, 14 1, 2 2 36 1, 2, 3, 4, 6, 9, 12, 18, 36 Step 2. Determine the GCF of the variables. The GCF of the variables is the one with the least exponent and is common to every term. 𝐺𝐶𝐹( 𝑥 3 𝑧 2 , 𝑥 2 𝑦 3 , 𝑦𝑧 4 ) = 1 Note that there are no factors common to all the three terms, this means that 𝑥 3 𝑧 2 , 𝑥 2 𝑦 3 , 𝑎𝑛𝑑 𝑦𝑧 4 are relatively prime. Hence, the GCF is 1. Step 3: Find the product of GCF of the numerical coefficient and the variables. (2)(1) = 2 Hence, 2 is the GCMF of 12𝑥 3 , 18𝑥𝑦, 24𝑥. Step 4. Find the other factor, by dividing each term of the polynomial 28𝑥 3 𝑧 2 − 14𝑥 2 𝑦 3 + 36𝑦𝑧 4 by the GCMF 2. 28𝑥 3 𝑧 2 14𝑥 2 𝑦 3 36𝑦𝑧 4 → − + 2 2 2 2 ∙ 14𝑥 3 𝑧 2 2 ∙ 7𝑥 2 𝑦 3 2 ∙ 18𝑦𝑧 4 → − + 2 2 2 → 14𝑥 3 𝑧 2 − 7𝑥 2 𝑦 3 + 18𝑦𝑧 4 Step 5: Write the complete factored form. 28𝑥 3 𝑧 2 − 14𝑥 2 𝑦 3 + 36𝑦𝑧 4 = 𝟐 (𝟏𝟒𝒙𝟑 𝒛𝟐 − 𝟕𝒙𝟐 𝒚𝟑 + 𝟏𝟖𝒚𝒛𝟒 ) 9 CO_Q1_Mathematics8_Module1A  Below is the summary of the steps of factoring the Greatest Common Monomial Factor. 1. Determine the number of terms. 2. Find the greatest common factor of the numerical coefficients. 3. Find the variable with the least exponent that appears in each term of the polynomial. It serves as the GCF of the variables. 4. Get the product of the greatest common factor of the numerical coefficient and the variables. It serves as the greatest common monomial factor of the given polynomial. 5. Find the other factor by dividing the given polynomial by its greatest common monomial factor and write the final factored form of the polynomial. What’s More Activity 1: Break the Great! Directions: Determine the Greatest Common Monomial Factor (GCMF) of each polynomial and write its factored form. Write your answer on a separate sheet of paper. Polynomial GCMF Factored Form 1. 𝑥2 + 2𝑥 2. 5𝑥 2 − 10𝑥 3 3. 25𝑥 2 𝑦 3 + 55𝑥𝑦 3 4. 10𝑐 3 − 80𝑐 5 − 5𝑐 6 + 5𝑐 7 5. 12𝑚5 𝑛2 − 6𝑚2 𝑛3 + 3𝑚𝑛 Questions: 1. How did you find the GCMF of the numerical coefficients of each term? 2. How did you find the GCMF of the variables in each term? 3. What did you do to the obtained GCF of the numerical coefficients and the GCF of the variables? 4. How did you find the remaining factors? 5. Did you have any difficulty in finding the GCF of the terms? 6. Did you have any difficulty in finding the remaining factor/s of polynomials after GCF is obtained? If so, why? If none, what helped you factor those expressions correctly? 10 CO_Q1_Mathematics8_Module1A Activity 2: You Complete Me Directions: Write a polynomial factor in the blank to complete each statement. Write the answers on your answer sheet. 1. 7𝑝 2 − 7𝑝 = 7𝑝 ( ______________) 2. 18𝑥𝑦 + 3𝑦 = ( ________________) (6𝑥 + 1) 3. 15𝑡 3 − 15𝑡 2 + 20𝑡 = 5𝑡 ( _________________________) 4. 17𝑥 5 − 51𝑥 4 − 34 𝑥 = ( ________________) (𝑥 4 − 3𝑥 3 − 2) 5. 35𝑥 5 𝑦 2 + 21𝑥 4 𝑦 + 14𝑥 3 𝑦 2 = 7𝑥 3 𝑦 ( _________________________) Questions: 1. Which was easier: finding the remaining factor given the GCF, or finding the GCF given the other factor? Why? 2. What did you do to find the GCF given the remaining factors? What I Have Learned Reflect on the activities you have done in this lesson by completing the following statements. Write your answers on your journal notebook. The 3 things that I have learned are ________________________________________________________ _________________________________________________________ _________________________________________________________ The one thing that I want to know more about is _________________________________________________________ 11 CO_Q1_Mathematics8_Module1A