CHAPTER R: Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 The Basics of Equation Solving R.6 Rational Expressions R.7 Radical Notation and Rational Exponents Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley R.4 Factoring Factor polynomials by removing a common factor. Factor polynomials by grouping. Factor trinomials of the type x2 + bx + c. Factor trinomials of the type ax2 + bx + c, a  1, using the FOIL method and the grouping method. Factor special products of polynomials. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Terms with Common Factors When factoring, we should always look first to factor out a factor that is common to all the terms. Example: 18 + 12x  6x2 = 6 • 3 + 6 • 2x  6 • x2 = 6(3 + 2x  x2) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Factoring by Grouping In some polynomials, pairs of terms have a common binomial factor that can be removed in the process called factoring by grouping. Example: x3 + 5x2  10x  50 = (x3 + 5x2) + (10x  50) = x2(x + 5)  10(x + 5) = (x2  10)(x + 5) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Trinomials of the Type x2 + bx + c Factor: x2 + 9x + 14. Solution: 1. Look for a common factor. 2. Find the factors of 14, whose sum is 9. Pairs of Factors Sum 1, 14 15 2, 7 9 The numbers we need. 3. The factorization is (x + 2)(x + 7). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Factor: 2y2  20y + 48. 1. First, we look for a common factor. 2(y2  10y + 24) 2. Look for two numbers whose product is 24 and whose sum is 10. Pairs Sum Pairs Sum 1, 24 25 2, 12 14 3, 8 11 4, 6 10 3. Complete the factorization: 2(y  4)(y  6). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Trinomials of the Type ax2 + bx + c, a  1 FOIL method 1. Factor out the largest common factor. 2. Find two First terms whose product is ax2. 3. Find two Last terms whose product is c. 4. Repeat steps (2) and (3) until a combination is found for which the sum of the Outside and Inside products is bx. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Factor: 8x2 + 10x + 3. (8x + )(x + ) (8x + 1)(x + 3) middle terms are wrong 24x + x = 25x (4x + )(2x + ) (4x + 1)(2x + 3) middle terms are wrong 12x + 2x = 14x (4x + 3)(2x + 1) Correct! 4x + 6x = 10x Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Grouping Method (or ac-method) ax2 + bx + c 1. Factor out the largest common factor. 2. Multiply the leading coefficient a and the constant c. 3. Try to factor the product ac so that the sum of the factors is b. That is, find integers p and q such that pq = ac and p + q = b. 4. Split the middle term. That is, write it as a sum using the factors found in step (3). 5. Factor by grouping. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Factor: 12a3  4a2  16a. 1. Factor out the largest common factor, 4a. 4a(3a2  a  4) 2. Multiply a and c: (3)(4) = 12. 3. Try to factor 12 so that the sum of the factors is the coefficient of the middle term, 1. (3)(4) = 12 and 3 + (4) = 1 4. Split the middle term using the numbers found in (3). 3a2 + 3a  4a  4 5. Factor by grouping. 3a2 + 3a  4a  4 = (3a2 + 3a) + (4a  4) = 3a(a + 1)  4(a + 1) = (3a  4)(a + 1) Be sure to include the common factor to get the complete factorization. 4a(3a  4)(a + 1) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Special Factorizations Difference of Squares A2  B2 = (A + B)(A  B) Example x2  25 = x2 – 52 = (x + 5)(x  5) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Special Factorizations Squares of Binomials A2 + 2AB + B2 = (A + B)2 A2  2AB + B2 = (A  B)2 Example x2 + 12x + 36 = x2 + 2(6) + 62 = (x + 6)2 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley More Factorizations Sum or Difference of Cubes A3 + B3 = (A + B)(A2  AB + B2) A3  B3 = (A  B)(A2 + AB + B2) Example 8y3 + 125 = (2y)3 + (5)3 = (2y + 5)(4y2  10y + 25) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley