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Lesson 5: Polynomials

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Part 1 of Factoring Polynomials

Russeneth Joy NaloFolgen

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- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 1 ST. MARY’S ACADEMY OF STA. CRUZ, INC. (FormerlyHolyCrossAcademy) Sta. Cruz, Davao del Sur FACTORING POLYNOMIALS GRADE 8 Lesson 1: Factoring Polynomials I Week # 1 Definition: Factoring is finding what to multiply to get an expression. Factoring a polynomial means writing it as a product of other polynomials. It is the reverse process of multiplication Examples 1. What are the possible dimension ofa rectangle whose area is 6 sq. cm and 16 sq.cm? A= 6 sq. cm A= 4 sq. cm Fig.1 L= 6 and w=1 L= 3 and w=2 Fig.2 L= 4 and w=1 L= 2 and w=2 2. What are the possible value ifthe product is 24? 24 x 1 = 24 6 x 4 = 24 Therefore,we can say that the possible factor of 24 are 1, 4, 6, and 24 Video Reference: Basic Concept of Factoring (https://www.youtube.com/watch?v=-VKAYqzRp4o) Visit this video The Common Monomial Factoring It is the reverse of multiplying a polynomial by a monomial. Howto Identify the Common Monomial Factor ofa Polynomial? 1. List down the factors of each terms 2. Identify the common terms present on both terms 3. After we have identified the common monomial factor, we will just get the remaining terms. My Learning Targets I can define factoring I can factorcompletely the polynomials with common monomial factor, difference of two squares; sum and difference of two cubes I can solve word probleminvolving factoring polynomials As you have noticed in the figures on the left, each figure has different values/ multipliers but it will still come up with same product. Fig1. The area is 6 sq. cm in it comes up with two different dimension which are 6:1 and 3:2 6:1 and 3:2 are what we called factor pairs Therefore,we can say that the factors of 6 are 1, 2, 3, 6 Fig2. The area is 8 sq. cm in it comes up with two different dimension which are 4:1 and 2:2 4:1 and 2:2 are factor pairs Therefore,we can say that the factors of 8 are 1, 2, and 4
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 2 4. In writing the factored form we will write first the common monomial and put parenthesis to remaining terms 5. Checked your solution Example showing step by step process: More Examples: Any polynomial that cannot be written as the product of two other polynomials except 1 and -1. Is said to be prime. A polynomial is said to be factored completely when it has been written as a product consisting only of prime factor. The following are some of the examples of prime: 5 11 x 𝟑𝒙 𝒙 + 𝟒 𝟐𝒙 − 𝟑 Example: What is the possible dimension of a rectangle whose areas is 𝒙 𝟐 − 𝟒𝒙 1. List down the factors of each terms 𝒙 𝟐 x ∙ x −𝟒 𝒙 -4 ∙ x 2. Identify the common terms present on both terms. In this example we have (x) this is the common monomial factor 3. Since we have identified the common monomial factor, we will just get the remaining terms. In this case, it is x and -4 the remaining terms 4. In writing the factored form we will write first the common monomial and put parenthesis to remaining terms In this case, we have 𝒙(𝒙 − 𝟒) this is now our factored Form 5. Checked your solution 𝒙 𝟐 − 𝟒 = 𝒙(𝒙 − 𝟒) 1. 4𝑥2 + 6𝑥 a. 4𝑥2= 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 6𝑥 = 2 ∙ 3 ∙ 𝑥 b. Common = 2 and x c. Remaining terms = 2x and 3 d. Factored form = 2x (2x+3) 2. 𝑚4 𝑛3 + 𝑚𝑛2 𝑚4 𝑛3 = 𝑚 ∙ 𝑚 ∙ 𝑚 ∙ 𝑚 ∙ 𝑛 ∙ 𝑛 ∙ 𝑛 𝑚𝑛2 = 𝑚 ∙ 𝑛 ∙ 𝑛 a. Common = m𝑛2 b. Remaining = 𝑚3 𝑛 and 1 c. m𝒏 𝟐(𝒎 𝟑 𝒏 + 1) Factored form Polynomials Common Monomial Factor Remaining Factor Factored Form a.) 2𝑥 + 10 b.) −8𝑚 – 12 c.) 6𝑥 – 12 d.) 4𝑎𝑦 – 24𝑎𝑧 2 −4 6 4𝑎 𝑥 + 5 2𝑚 + 3 𝑥 – 2 𝑦 – 6𝑧 2(𝑥 + 5) −4(2𝑚 + 3) 6(𝑥 − 2) 4𝑎(𝑦 − 6𝑧) Standard Form Factored Form
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 3 Factoring by Grouping The use of Factoring by Grouping is that some expression doesn’t have common monomial factor but instead it has some polynomial expressions that have common binomial factor. Video Reference on Factoring by Grouping: (https://www.youtube.com/watch?v=NYMpf4fdnbY) Factoring the Difference of Two Squares If x and y are real numbers, variables, and algebraic expression, then General Formula: 𝒙 𝟐 − 𝒚 𝟐 = (𝒙 + 𝒚)(𝒙 − 𝒚) Examples: If you encounter this kind of expression this is the steps. Example 1: 𝒙𝒚+ 𝟑𝒙 + 𝟔𝒚 + 𝟏𝟖 a. Divide the terms by two In this expression : 𝑥𝑦 + 3𝑥 + 6𝑦 + 18. We have: 𝑥𝑦 + 3𝑥 6𝑦 + 18 b. Factor the divided terms and find the common monomial factor 𝑥𝑦 + 3𝑥 6𝑦 + 18 Common Monomial Factor: =x = 6 Remaining Term: (𝑦 + 3) (𝑦 + 3) So we have: 𝑥(𝑦 + 3) and 6(𝑦 + 3). The Common Binomial Factor is (𝒚 + 𝟑) c. Add the remaining terms and multiply it to the common monomial factor In the expression: 𝑥(𝑦 + 3) + 6(𝑦 + 3). The Common Monomial Factoris (𝑦 + 3) The Remaining Termis x and 6. So if we add it. It becomes (𝒙 + 𝟔) d. The factored form is (𝒚 + 𝟑)(𝒙 + 𝟔) e. Checked your solution. If your answer is the same to the given, then it is correct but if not please go back to your solution. Check Using FOIL METHOD (𝑦 + 3)(𝑥 + 6) = 𝑥( 𝑦) + 𝑥(3) + 6( 𝑦) + 6(3) = 𝑥𝑦 + 3𝑥 + 6𝑦 + 18 Example 2. ( 𝒂 + 𝒃)( 𝒂 − 𝟑) + (𝒂 + 𝒃)(𝒂− 𝟒) In this example you have noticed it is different to the other one. But you also have noticed that the common binomial factor is already given. So here are the steps; a. Identify the common binomial factor and remaining terms Common Binomial Factor = ( 𝑎 + 𝑏) Remaining Terms = ( 𝑎 − 3) and (𝑎 − 4) b. To have a factored form just add the remaining terms and multiply it to the common binomial term ( 𝑎 + 𝑏)[ ( 𝑎 − 3) + (𝑎 − 4)] ( 𝑎 + 𝑏)( 𝑎 + 𝑎 − 3 − 4) ( 𝑎 + 𝑏)(2𝑎 − 7) Simplify the final answer if needed. In this case, we have to simplify it FINAL ANSWER: ( 𝒂 + 𝒃)(𝟐𝒂 − 𝟕)
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 4 In other words: the difference of the squaresof two terms is the product of the sumand difference of those terms. A polynomial is a difference of two squares if it satisfies these three conditions: (a) It is a binomial and each term is a square; (b) There is a minus sign between the two terms; (c)The operation involved is subtraction Steps on how to factored the difference of two square. 1. Find the Square root of the terms 2. Substitute in the pattern or the general formula Example 1: 𝒙 𝟐 − 𝟗 1. Find the Square root of the terms. (Review on square root) √𝑥2 = x since: x∙x = 𝑥2 √9 = 3 since: 3∙3 = 9 2. Substitute in the pattern 𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦) 𝒙 𝟐 − 𝟗 𝟐 = ( 𝒙 + 𝟑)( 𝒙 − 𝟑) Example 2: 𝟒𝒙 𝟐 − 𝟐𝟓𝒚 𝟐 1st step √4𝑥2= 2x √25𝑦2 = 5y 2nd step 𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦) 𝟒𝒙 𝟐 − 𝟐𝟓𝒚 𝟐 = (𝟐𝒙 + 𝟓𝒚)(𝟐𝒙 − 𝟓𝒚) Factoring the Sum or Difference of Two Cubes General Formula: (𝒂 𝟑 + 𝒃 𝟑) = (𝒂 + 𝒃)(𝒂 𝟐 − 𝒂𝒃 + 𝒃 𝟐) (𝒂 𝟑 − 𝒃 𝟑) = (𝒂 − 𝒃)(𝒂 𝟐 + 𝒂𝒃 + 𝒃 𝟐) Note: Please observe carefully and focus on the sign A polynomial is a sum or difference of two cubes if its satisfies these condition: (a) It is binomial; (b)Each term is a cube Steps on how to factored the sum or difference of two cube 1. Look at the given and identify what general formula is applicable. 2. Identify the cube root of the two terms. 3. In writing the factored form we should be guided to general formula General Formula: (𝑎3 + 𝑏3 ) = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ) (𝑎3 − 𝑏3 ) = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2 ) 4. Check your solution. More Example: 𝒙 𝟑 + 𝟐𝟕 Pattern : (𝑎3 + 𝑏3) = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) Cube Root: 𝒙 𝟑 = (𝒙) 𝟑 = 𝑥 ∙ 𝑥 ∙ 𝑥 𝟖 = (𝟑) 𝟑= 3 ∙ 3 ∙ 3 Substitute: 𝒂 = 𝒙 𝒂𝒏𝒅 𝒃 = 𝟑
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 5 (𝑎3 + 𝑏3) = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) (𝑥3 + 33) = (𝑥 + 3)(𝑥2 − 3𝑥 + 32) (𝑥3 + 27) = (𝒙 + 𝟑)(𝒙 𝟐 − 𝟑𝒙 + 𝟗) Word Problems involving Factoring Polynomials ST. MARY’S ACADEMY OF STA. CRUZ, INC. (FormerlyHolyCrossAcademy) Sta. Cruz, Davao del Sur MATHEMATICS 8 Name: _______________________________ Date: _________________ Grade Level & Section: ___________________ Week # 1 Topic: Factoring Polynomials Example : 𝒙 𝟑 − 𝟖 1. Look at the given and identify what general formula is applicable. In this example, it is applicable to (𝒂 𝟑 − 𝒃 𝟑 ) = (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃 𝟐 ) (𝒂 𝟑 − 𝒃 𝟑 ) this is the Standard Form (𝒂 − 𝒃)(𝒂 𝟐 + 𝒂𝒃 + 𝒃 𝟐 ) this is the Factored Form 2. Identify the cube root 𝒙 𝟑 = (𝒙) 𝟑 = 𝑥 ∙ 𝑥 ∙ 𝑥 𝟖 = (𝟐) 𝟑 = 2 ∙ 2 ∙ 2 3. In writing the factored form we should be guided to general formula In this example we use: ( 𝒂 − 𝒃)(𝒂 𝟐 + 𝒂𝒃 + 𝒃 𝟐 ) we just follow the formula 𝒂 = 𝒙 = ( 𝒙 − 𝟐)(𝒙 𝟐 + 𝟐𝒙 + 𝟐 𝟐 ) 𝒃 = 𝟐 = ( 𝒙 − 𝟐)(𝒙 𝟐 + 𝟐𝒙 + 𝟒 ) this is now the factored form 4. Check your solution. References: Nivera, Gladys. (2018). Grade 8 Mathematics: Patterns and Practicalities. SalesianaBooks by Don Posco Press,Inc. Orance,O.,Mendoza, Marilyn. (2019). E-Math: Worktext in Mathematics. Rex Book Store, Inc.
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 6 Note: If you have questions about our lesson, ask the person the has more knowledge to guide you like your older sibling or your parents. Activity No1. (Common Monomial and Binomial Factor) Direction: Factor the following polynomials. Identify the common factors, remaining factors and write the factored form. Present this on a table and write your answer on an intermediate paper. Show your solution and observe the proper steps in solving. Activity No 2: (Sum and Difference of Two Cubes and Difference of Two Square) Direction. Factor the Following Polynomials. Write your answer on an intermediate paper. 1. 𝑥2 − 49 6. 𝑥3 − 8 2. 𝑚2 − 36 7. 8𝑥3 − 1 3. 𝑟2 − 9 8. 𝑦3 + 64 Activity No 3: (Word Problem) Direction: Solve the following word problem. Show your solution and observe the proper steps in solving. Write your answer on an intermediate paper. 1. The side of square cloth measures 2𝑥 + 5. A small square is cut from the larger square as shown. Write an expression for the remaining area in factored form. (the figure is on the right) Assessment: Please help us make our module a learner-friendly tool by answering the following questions. Write your comments on this paper. 1. What aspects of this module are most helpful to your learning? 1. 𝟑𝒙 + 𝟔 2. 𝟏𝟓𝒙 𝟐 𝒚 𝟑 − 𝟑𝟑𝒙𝒚 𝟑 3. 𝟓𝒙 + 𝟏𝟎 4. 25𝒙 𝟐 𝒚 𝟑 − 𝟓𝟓𝒙𝒚 𝟑 5. 𝟏𝟐𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟔 𝑥 − 2 2𝑥 + 2
- ST. MARY’S ACADEMY OF STA. CRUZ, INC. 7 2. What aspects of this module are least helpful to your learning? 3. How should we improve the module for you? Prepared by: Checked by: MA. RUSSENTH JOY N. NALO,LPT KRIS GIA T. ESCUETA, LPT Teacher Academic Coordinator Verified/Noted by: Approved by: S. MARIA RUFEL S. PALARCA,RVM S. MA. YOLANDA D. CAPIÑA,RVM Assistant School Principal School Principal

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