Factoring polynomials with common monomial factor
- 1. Prepared by: Gauben L. Malicsi Factoring Out the Greatest Common Factor (GCF) A natural number larger than 1 that has no factors other than itself and 1 is called prime number. The numbers, 2, 3, 5, 7, 11, 13, 17, 19, 23 are the first nine prime numbers. There are infinitely many prime numbers. To factor a natural numbers completely means to write it as a product of prime numbers. In factoring 12, we might write 12 = 4 x 3. However, 12 is not factored completely as 4 x 3 because 4 is not a prime. To factor completely, we write 12 = 2 x 2 x 3. The greatest common factor (GCF) is a monomial that includes every number or variable that is a factor of all terms of the polynomial. We can use the following strategy for finding the greatest common factor of a group of terms. Strategy for Finding the Greatest Common Factor (GCF) 1. Find the greatest common factor of the numerical coefficients. 2. Find the variable with the least exponent that appears in each term of the polynomial. 3. The product of the greatest common factor in (a) and (b) is the GCF of the polynomial. 4. To completely factor the given polynomial, divide the polynomial by its GCF, the resulting quotient is the other factor. EXAMPLE 1: Factoring out the greatest common factor Factor each polynomial by factoring out the GCF. a. 5𝑥4 − 10𝑥3 + 15𝑥2 b. 8𝑥𝑦2 + 20𝑥2 𝑦 SOLUTIONS: a. 5𝑥4 − 10𝑥3 + 15𝑥2 Find the GCF of the numerical coefficients. 5 = 5 x 1 10 = 5 x 2 15 = 5 x 3 The common factor is 5. So, the GCF for numerical coefficients is 5. Find the variable with the least exponent that appears in each term of the polynomial. 𝑥4 𝑥3 𝑥2 The least exponent for x is 2. Therefore, the GCF for variable is 𝑥2 Find the product of (a) and (b). (5)(𝑥2 ) = 5𝑥2 5𝑥2 is now the GCF of the polynomial Divide the polynomial by its GCF, the resulting quotient is the other factor. 5𝑥4 − 10𝑥3 + 15𝑥2 = 5𝑥2 (𝑥2 − 2𝑥 + 3) b. 8𝑥𝑦2 + 20𝑥2 𝑦 Find the greatest common factor of the numerical coefficients. 8 = 4 x 2 20 = 4 x 5 The common factor is 4. So, the GCF for numerical coefficients is 4. Find the variable with the least exponent that appears in each term of the polynomial. 𝑥𝑦2 𝑥2 𝑦 Factoring polynomials with common monomial factor Factors completely polynomials with common monomial factor COMPETENCY 1.1
- 2. Prepared by: Gauben L. Malicsi The least exponent for both x and y is 1. Therefore, the GCF for variables is 𝑥𝑦. Find the product of (a) and (b). (4)(xy) = 4xy 4xy is now the GCF of the polynomial Divide the polynomial by its GCF, the resulting quotient is the other factor. 8𝑥𝑦2 + 20𝑥2 𝑦 = 4𝑥𝑦(2𝑦 + 5𝑥) EXAMPLE 2: Factoring out binomial Factor. a. (x + 3)w + (x + 3)a b. x(x – 9) – 4(x – 9) SOLUTIONS: a. We treat x + 3 like a common monomial when factoring: (x + 3)w + (x + 3)a = (x + 3)(w + a) b. Factor out the common binomial x – 9: x(x – 9) – 4(x – 9) = (x – 4)(x – 9) PERFORMANCE TASK Direction: Complete the table below. Polynomial Greatest Common Monomial Factor (CMF) Quotient of Polynomial and CMF Factored Form 6m + 8 2 3m + 4 2(3m+4) 4𝑚𝑛2 4𝑚𝑛2 (3𝑚 + 𝑛) 18𝑑2 𝑜3 𝑡6 − 15𝑑6 𝑜4 6𝑡6 − 5𝑑4 4(12) + 4(8) 4 12𝑤𝑖3 𝑛5 − 16𝑤𝑖𝑛 + 20𝑤𝑖𝑛𝑛𝑒𝑟 ASSESSMENT TASK Show your solutions on the box provided below and write your final answer on the space before the number. 1. Factor out the greatest common factor in each expression. (See Example 1) a. 𝑥3 − 5𝑥 b. 10𝑥2 − 20𝑦3 c. 42𝑤𝑧 + 28𝑤𝑎 2. Factor out the greatest common factor in each expression. (See Example 2) a. (x – 6)a + (x – 6)b b. w – 2)y +(w – 2)3 YOUR SOLUTIONS: 1. 2. 3. 4. 5.
- 3. Prepared by: Gauben L. Malicsi INDICATORS Meets Standard of Excellence Approaching Standard of Excellence Meets Acceptable Standard Does Not Meet Acceptable Standard CRITERIA 4 3 2 1 Performance Task Shows exemplary performance Demonstrates solid performance and understanding With some errors and mastery is not thorough. Has errors, omission and misconception. Assessment Task With 5 correct answers With 4 correct answers With 3 correct answers With less than 3 correct answers Completeness Has all aspects of work that exceed level of expectation Has some aspects of work that exceed level of expectation Has minimal aspects of work that meet level of expectation No aspect of work meets level of expectations Neatness The learning tasks are done very neatly. The learning tasks are done neatly. The learning tasks are done quite neatly. The learning tasks are poorly done and need improvement. Submission of Requirements The assigned learning tasks are submitted on or before the deadline. The assigned learning tasks are submitted a day after the deadline. The assigned learning tasks are submitted two days after the deadline. The assigned learning tasks are submitted three days after the deadline. TOTAL SCORE _________________________________ _________________________________ Parent/Guardian’s Signature Over Printed Name Student’s Signature Over Printed Name