Transcript Chapter 5-4
Chapter 5 Polynomials and Polynomial Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1 1 Chapter Sections 5.1 – Addition and Subtraction of Polynomials 5.2 – Multiplication of Polynomials 5.3 – Division of Polynomials and Synthetic Division 5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping 5.5 – Factoring Trinomials 5.6 – Special Factoring Formulas 5.7-A General Review of Factoring 5.8- Polynomial Equations Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-2 2 § 5.4 Factoring a Monomial from a Polynomial and Factoring by Grouping Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-3 3 Factors A prime number is an integer greater than 1 that has exactly two factors, 1 and itself. A composite number is a positive integer that is not prime. Prime factorization is used to write a number as a product of its primes. 24 = 2 · 2 · 2 · 3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-4 4 Factors To factor an expression means to write the expression as a product of its factors. If a · b = c, then a and b are factors of c. a·b Recall that the greatest common factor (GCF) of two or more numbers is the greatest number that will divide (without remainder) into all the numbers. Example: The GCF of 27 and 45 is 9. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-5 5 Determining the GCF 1. Write each number as a product of prime factors. 2. Determine the prime factors common to all the numbers. 3. Multiply the common factors found in step. The product of these factors is the GCF. Example: Determine the GCF of 24 and 30. 24 = 2 · 2 · 2 · 3 30 = 2 · 3 · 5 A factor of 2 and a factor of 3 are common to both, therefore 2 · 3 = 6 is the GCF. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-6 6 Determining the GCF To determine the GCF of two or more terms, take each factor the largest number of times it appears in all of the terms. Example: 12 4 9 7 a.) y , y , y , y . Note that y4 is the highest power of y common to all four terms. The GCF is, therefore, y4. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-7 7 Factoring a Monomial from a Polynomial 1. Determine the GCF of all the terms in the polynomial. 2. Write each term as the product of the GCF and another factor. 3. Use the distributive property to factor out the GCF. Example: 15x4 – 5x3+25x2 (GCF is 5x2) 15 x 5 x 25 x 5 x 3 x 5 x x 5 x 5 4 3 2 2 2 2 2 5 x (3 x x 5) 2 2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-8 8 Factor a Common Binomial Factor Sometimes factoring involves factoring a binomial as the greatest common factor. Example Factor 3 x(5 x 6) 4(5 x 6). The GCF is 5 x 6. Factoring out the GCF gives 3x(5 x 6) 4(5 x 6) (5 x 6)(3x 4) Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-9 9 Factoring by Grouping The process of factoring a polynomial containing four or more terms by removing common factors from groups of terms is called factoring by grouping. Example: Factor x2 + 7x + 3x + 21. x(x + 7) + 3(x + 7) = (x + 7) (x + 3) Use the FOIL method to check your answer. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-10 10 Factor by Grouping Method 1. Determine if all four terms have a common factor. If so, factor out the greatest common factor from each term. 2. Arrange the four terms into two groups of two terms each. Each group of two terms must have a GCF. 3. Factor the GCF from each group of two terms 4. If the two terms formed in step 3 have a GCF, factor it out. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-11 11 Factor by Grouping Method Example: Factor x3 -5x2 + 2x - 10. There are no factors common to all four terms. However, x2 is common to the first two terms and 2 is common to the last two terms. Factor x2 from the first two terms and factor 2 from the last two terms. x 5 x 2 x 10 x ( x 5) 2( x 5) 3 2 2 ( x 5)( x 2) 2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-12 12 Factoring by Grouping Example: a.) Factor by grouping: x3 + 2x + 5x2 – 10 There are no factors common to all four terms. Factor x from the first two terms and -5 from the last two terms. x 2 x 5 x 10 x( x 2) 5( x 2) 3 2 2 2 ( x 2)( x 5) 2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-13 13