Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 2 Factoring and Applications Chapter 6

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 4 Objectives 1.Find the greatest common factor of a list of numbers. 2.Find the greatest common factor of a list of variable terms. 3.Factor out the greatest common factor. 4.Factor by grouping. 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 5 Finding the Greatest Common Factor of a List of Numbers The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. This means 6 is the greatest common factor of 18 and 24, since it is the largest of their common factors. Note Factors of a number are also divisors of the number. The greatest common factor is the same as the greatest common divisor. 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 6 (a) 36, 60 Example 1 Find the greatest common factor for each list of numbers. First write each number in prime factored form. Finding the Greatest Common Factor of a List of Numbers 36 = 2 · 2 · 3 · 360 = 2 · 2 · 3 · 5 Use each prime the least number of times it appears in all the factored forms. Here, the factored forms share two 2’s and one 3. Thus, GCF = 2 · 2 · 3 = Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 7 (b) 18, 90, 126 Example 1 (continued) Find the greatest common factor for each list of numbers. Find the prime factored form of each number. Finding the Greatest Common Factor of a List of Numbers 18 = 2 · 3 · 390 = 2 · 3 · 3 · 5 All factored forms share one 2 and two 3’s. Thus, GCF = 2 · 3 · 3 = = 2 · 3 · 3 · Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 8 (c) 48, 61, 72 Example 1 (concluded) Find the greatest common factor for each list of numbers. 48 = 2 · 2 · 2 · 2 · 361 = 1 · 61 There are no primes common to all three numbers, so the GCF is 1. GCF = 1 72 = 2 · 2 · 2 · 3 · 3 Finding the Greatest Common Factor of a List of Numbers 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 9 (a) 12x 2, –30x 5 Example 2 Find the greatest common factor for each list of terms. 12x 2 = 2 · 2 · 3 · x 2 First, 6 is the GCF of 12 and –30. The least exponent on x is 2 (x 5 = x 2 · x 3 ). Thus, GCF = 6x 2. –30x 5 = –1 · 2 · 3 · 5 · x 5 Finding the Greatest Common Factor for Variable Terms Note The exponent on a variable in the GCF is the least exponent that appears on that variable in all the terms. 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 10 Note In a list of negative terms, sometimes a negative common factor is preferable (even though it is not the greatest common factor). In (b) above, we might prefer –x 4 as the common factor. (b) –x 5 y 2, –x 4 y 3, –x 8 y 6, –x 7 Example 2 (concluded) Find the greatest common factor for each list of terms. There is no y in the last term. So, y will not appear in the GCF. There is an x in each term, and 4 is the least exponent on x. Thus, GCF = x 4. –x 5 y 2, –x 4 y 3, –x 8 y 6, –x 7 Finding the Greatest Common Factor for Variable Terms 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 11 Finding the Greatest Common Factor (GCF) Step 1Factor. Write each number in prime factored form. Step 2List common factors. List each prime number or each variable that is a factor of every term in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.) Step 3Choose least exponents. Use as exponents on the common prime factors the least exponents from the prime factored forms. Step 4Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1. Finding the Greatest Common Factor for Variable Terms 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 12 CAUTION The polynomial 3m + 12 is not in factored form when written as the sum 3 · m + 3 · 4. Not in factored form The terms are factored, but the polynomial is not. The factored form of 3m + 12 is the product 3(m + 4). In factored form Factor Out the Greatest Common Factor 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 13 (a) 24x 5 – 40x 3 Example 3 Factor out the greatest common factor. Factor Out the Greatest Common Factor GCF = 8x 3 = 8x 3 (3x 2 – 5) = 8x 3 (3x 2 ) – 8x 3 (5) Note If the terms inside the parentheses still have a common factor, then you did not factor out the greatest common factor in the previous step. 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 14 Example 3 (concluded) Factor out the greatest common factor. Factor Out the Greatest Common Factor CAUTION Be sure to include the 1 in a problem like Example 3(b). Check that the factored form can be multiplied out to give the original polynomial. 6.1 Factors; The Greatest Common Factor (b) 4x 6 y 4 – 20x 4 y 3 + x 2 y 2 = x 2 y 2 (4x 4 y 2 ) – x 2 y 2 (20x 2 y) + x 2 y 2 (1) = x 2 y 2 (4x 4 y 2 – 20x 2 y +1)

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 15 – 3x 5 – 15x 3 + 6x 2 Example 4 Factor – 3x 5 – 15x 3 + 6x 2. Factor Out the Greatest Common Factor GCF = – 3x 2 = – 3x 2 (x 3 + 5x – 2) Note Whenever we factor a polynomial in which the coefficient of the first term is negative, we will factor out the negative common factor, even if it is just – Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 16 Example 5 Factor out the greatest common factor. Factor Out the Greatest Common Factor w 2 (z 4 – 3) + 5(z 4 – 3) Here, the binomial z 4 – 3 is the GCF. w 2 (z 4 – 3) + 5(z 4 – 3)= (z 4 – 3)(w 2 + 5) 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 17 6x + 4xy – 10y – 15 Example 6 Factor by grouping. Factor By Grouping If we leave the terms grouped as they are, we could try factoring out the GCF from each pair of terms. 6x + 4xy – 10y – 15= 2x(3 + 2y) – 5(2y + 3) This works, showing a common binomial of 2y + 3 in each term. 6x + 4xy – 10y – 15= 2x(2y + 3) – 5(2y + 3) = (2y + 3)(2x – 5) 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 18 CAUTION Be careful with signs when grouping in a problem like Example 6. It is wise to check the factoring in the second step before continuing. Factor Out the Greatest Common Factor 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 19 Factor By Grouping Factoring a Polynomial with Four Terms by Grouping Step 1Group terms. Collect the terms into two groups so that each group has a common factor. Step 2Factor within groups. Factor out the greatest common factor from each group. Step 3Factor the entire polynomial. Factor a common binomial factor from the results of Step 2. Step 4If necessary, rearrange terms. If Step 2 does not result in a common binomial factor, try a different grouping. 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 20 10a 2 – 12b + 15a – 8ab Example 7 Factor by grouping. Factor By Grouping Working as before, we get This does not work. These two factored terms have no binomial in common. So, we will group another way. 10a 2 – 12b + 15a – 8ab= 2(5a 2 – 6b) + a(15 – 8b) 6.1 Factors; The Greatest Common Factor

Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.1 – Slide 21 Factor By Grouping Example 7 (concluded) Factor by grouping. This works, showing a common binomial of 5a – 4b in each term. Thus, 10a 2 – 12b + 15a – 8ab= (5a – 4b)(2a + 3) 6.1 Factors; The Greatest Common Factor 10a 2 – 12b + 15a – 8ab= 10a 2 – 8ab + 15a – 12b = 2a(5a – 4b) + 3(5a – 4b)