1. OBJECTIVES 5.1 Introduction to Factoring Slide 1Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aFind the greatest common factor, the GCF, of monomials. bFactor polynomials when the terms have a common factor, factoring out the greatest common factor. cFactor certain expressions with four terms using factoring by grouping.

2. 5.1 Introduction to Factoring Multiply Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 3 (a + 1) =3a + 3 Factor Factor = Product Factor Factor = Polynomial

3. 5.1 Introduction to Factoring Factoring Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. To factor a polynomial is to find an equivalent expression that is a product. Since we know: Factor Factor = Polynomial 3 (a + 1) = 3a + 3 We have the factors of 3a + 3.

4. 5.1 Introduction to Factoring Factoring Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. So, 3 is a factor of 3a +3 and (a + 1) is a factor of 3a + 3 polynomial = factor factor 3a + 3 = 3 (a +1) An equivalent expression of this type is called a factorization of the polynomial.

5. 20 and 30 have several factors in common. The largest of these common factors is the greatest common factor, GCF. One way to find the GCF is by making a list of the factors of each number. 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. (continued) Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

6. The factors of 20: 1, 2, 4, 5, 10, and 20 The factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30 Common numbers: 1, 2, 5, and 10. The GCF is 10. 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. (continued) Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

7. Another way to find the GCF. Find the prime factorization of each number. Then draw lines between common factors. 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

8. EXAMPLE Solution 20 = 2 2 5 30 = 2 3 5 Draw lines between the common factors. The GCF is 10. 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. AFind the GCF of 20 and 30. Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

9. EXAMPLE Solution : Write the prime factorization of each number. 420 = 2 · 2 · 3 · 5 · 7 924 = 2 · 2 · 3 · 7 · 11 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. BFind the GCF of 420 and 924. Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. So the GCF is 2 · 2 · 3 · 7 = 84

10. To factor a polynomial is to express it as a product. factor factor A factorization of a polynomial is an expression that names that polynomial as a product. 5.1 Introduction to Factoring Factor; Factorization Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

11. EXAMPLE x · x · x x · x · x · x Solution Prime factorization of each coefficient: 30x 3 = 2 · 3 · 5 · –48x 4 = –1 · 2 · 2 · 2 · 2 · 3 · The GCF of the coefficients is 6. 5.1 Introduction to Factoring a Find the greatest common factor, the GCF, of monomials. CFind the GCF of 30x 3, –48x 4, 54x 5, and 12x 2. Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The GCF of these monomials is x 3, because 3 is the smallest exponent of x. The GCF is 6x 3. x3x4x3x4

12. 1.Find the prime factorization of the coefficients, including –1 as a factor if any coefficient is negative. 2. Determine any common prime factors of the coefficients. For each one that occurs, include it as a factor of the GCF. If none occurs, use 1 as a factor. 5.1 Introduction to Factoring To find the GCF of Two or more Monomials (continued) Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

13. 3.If any variable appears as a factor of all the monomials, include it as a factor, using the smallest exponent of the variable. If novariable occurs in all the monomials, use 1 as a factor. 4. The GCF is the product of the results of steps (2) and (3). 5.1 Introduction to Factoring To find the GCF of Two or more Monomials (continued) Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

14. EXAMPLE Solution 9a  21 = (3 · 3 · a) - (3 · 7 ) Factoring each term = 3(3a – 7) Factoring out the GCF Check: 3(3a  7) = 9a  21 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. D Factor: 9a  21. Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

15. EXAMPLE Solution 28x 6 + 32x 3 = (4 · 7 · ) + (4 · 8 · ) Factoring each term = 4x 3 (7x 3 +8) Factoring out the GCF 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. EFactor: 28x 6 + 32x 3. Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. x 6 x 3

16. Factoring When Terms Have a Common Factor To factor a polynomial with two or more terms of the form ab + ac, we use the distributive law with the sides of the equation switched: ab + ac = a(b + c). MultiplyFactor 4x(x 2 + 3x – 4) 4x 3 + 12x 2 – 16x 4x·x 2 + 4x·3x – 4x·4 4x·x 2 + 4x·3x – 4x·4 4x 3 + 12x 2 – 16x 4x(x 2 + 3x – 4) 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

17. EXAMPLE What is the largest common constant factor? What is the largest common variable factor? Factor out 3x 3 12x 5  21x 4 + 24x 3 3x 3 (4x 2  7x + 8) 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. F Factor: 12x 5  21x 4 + 24x 3 Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 3x33x3

18. EXAMPLE Solution 9a 3 b 4 + 18a 2 b 3 9 a 2 b 3 (ab + 2) The largest common factor is 9a 2 b 3. 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. GFactor: 9a 3 b 4 + 18a 2 b 3 Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

19. EXAMPLE Solution  4xy + 8xw  12x =  4x(y  2w + 3) 5.1 Introduction to Factoring b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. H Factor:  4xy + 8xw  12x Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

20. STUDY TIP Tips for Factoring Before doing any other kind of factoring, first try to factor out the GCF. Always check the result of factoring by multiplying. Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

21. Factoring by Grouping Sometimes algebraic expressions contain a common factor with two or more terms. 5.1 Introduction to Factoring c Factor certain expressions with four terms using factoring by grouping. Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

22. EXAMPLE Solution: The binomial (x + 2) is a factor of both x 2 (x + 2) and 3(x + 2). Thus, x + 2 is a common factor. x 2 (x + 2) + 3(x + 2) (x 2 + 3) (x + 2) The factorization is (x 2 + 3)(x + 2). 5.1 Introduction to Factoring c Factor certain expressions with four terms using factoring by grouping. IFactor x 2 (x + 2) + 3(x + 2). Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

23. If a polynomial can be split into groups of terms and the groups share a common factor, then the original polynomial can be factored. This method, known as factoring by grouping, can be tried on any polynomial with four or more terms. 5.1 Introduction to Factoring c Factor certain expressions with four terms using factoring by grouping. Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

24. EXAMPLE a) 3x 3 + 9x 2 + x + 3 = (3x 3 + 9x 2 ) + (x + 3) Don’t forget to include the 1. 5.1 Introduction to Factoring c Factor certain expressions with four terms using factoring by grouping. JFactor by grouping. (continued) Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. = 3x23x2 + 1 (x + 3)

25. EXAMPLE b) 9x 4 + 6x  27x 3  18 (9x 4 + 6x) + (  27x 3  18) 3x(3x 3 + 2) + (  9)(3x 3 + 2) (3x – 9)(3x 3 + 2) 5.1 Introduction to Factoring c Factor certain expressions with four terms using factoring by grouping. JFactor by grouping Slide 25Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.