1. Beginning Algebra 5.1 The Greatest Common Factor and Factoring by Grouping

2. 5.1 Factor by Grouping Objective 1. To factor the greatest common factor from a polynomial. Objective 2. To factor by grouping.

3. Some rules governing factoring in elementary algebra: 2. The powers must be whole numbers. 1. The numbers (coefficients) must be real real and rational rational. Factoring Polynomials: 3. Prime factors will be binomials binomials or polynomials that cannot be factored further in real real numbers.

4. b) The sum of an even degree term (2 or higher) and any real number, such as: (x2 (x2 + 4), (x2y2 (x2y2 + 9), (2x 2 + 25), (x (x 4 + 3), etc etc. Prime factors Prime factors include: a) Linear polynomials polynomials, such as: (x + 3), (x + y + z)z) c) The sum of two even-powered terms terms, such as: (x (x 4 + y 4 ), (x6 (x6 + y 8 ), (x2 (x2 + y6)y6) Factoring Polynomials: d) Certain quadratic trinomials trinomials which cannot be expressed as the product product of two linear binomials with rational numbers numbers.

5. Factor Common Factors: a) 3x 2 + 12 3(x2 3(x2 + 4)4) prime prime factors b) 5x 3 + 7x 2 x 2 (5x + 7)7) prime factors 3·x2 3·x2 + 3·43·4 x 2 ·5x + x2·7x2·7 What to DoHow to Do It 1. Factor out the common factor(s) from each term. distributive property 2. Apply the distributive property. common factors 3. Integers as common factors are left in composite form. common factors 4. Letters as common factors are left in power form.

6. What to DoHow to Do It c) 3x 3  21x 2 + 27x d) 4x 2  20x + 24 4(x 2  5x + 6) 3x(x 2  7x + 9) 3x·x 2  3x·7x + 3x·9 4·x 2  4·5x + 4·6 1. Factor out the common factor(s) from each term. distributive property 2. Apply the distributive property. Factor Common Factors: common factors 3. Integers as common factors are left in composite form. common factors 4. Letters as common factors are left in power form. prime factors not a prime factor Some polynomials polynomials factor further:.

7. x 2  3x  2x + 6 Factor by Grouping: What to DoHow to Do It x 2  3x  2x + 6 (x  3)(x  2)2) x(x  3) 2(x  3)3) given Group the given polynomial's 4 terms Bring down middle sign first twolast two by underlining the first two and last two. Factor common factor factor from each group group. Factor common factor factor from each group. x  3 Now the common factor is (x  3). Underline it. x(x  3)  2(x  3)

8. Check Factors by FOIL What to DoHow to Do It  Check by multiplying back out by First Note the sum of O + I terms F 0 I L (x  3)(x  2) x 2  5x + 6 Outer Inner Last x 2  2x  3x + 6 x2 x2 6  2x  2x  3x x 2  5x + 6 x 2  3x  2x + 6 

9. x 2 + 7x + 2x + 14 Factor by Grouping: What to DoHow to Do It x 2 + 7x + 2x + 14 (x + 7)(x + 2)2) x(x +7) 2(x + 7)7) given Group the given polynomial's 4 terms Bring down middle sign first twolast two by underlining the first two and last two. Factor common factor factor from each group group. Factor common factor factor from each group. x + 7 Now the common factor is (x + 7). Underline it. x(x + 7) + 2(x + 7) +

10. Check Factors by FOIL  Check by multiplying multiplying by First F 0 IL Outer Inner Last Note sum of O + I terms: x 2 + 7x + 2x + 14  x 2 + 2x + 7x + 14 x2 x2 14 + 2x + 2x + 7x x 2 + 9x + 14 What to DoHow to Do It (x + 7)(x + 2) x 2 + 9x + 14

11. 3x 2  9x + 2x  6 Factor by Grouping: What to DoHow to Do It 3x 2  9x + 2x  6 (x  3)(3x + 2)2) 3x(x  3) 2(x  3)3) given Group the given polynomial's 4 terms Bring down middle sign first twolast two by underlining the first two and last two. Factor common factor factor from each group group. Factor common factor factor from each group. x  3 Now the common factor is (x  3). Underline it. 3x(x  3) + 2(x  3) +

12. Check Factors by FOIL What to DoHow to Do It Note sum of O + I terms: (3x + 2)(x  3) 3x 2  7x  6 3x 2  9x + 2x  6 3x 2  6  9x +2x 3x 2 + 2x  9x  6  3x 2  7x  6  Check by multiplying multiplying by First F 0 IL Outer Inner Last

13. Factor by Grouping: 4x 2 (2x  3) 7(2x  3)3) 8x 3  12x 2 + 14x  21 (4x 2 + 7)(2x  3) PRIME 4x 2 (2x  3) + 7(2x  3) given Group the given polynomial's 4 terms Bring down middle sign first twolast two by underlining the first two and last two. Factor common factor factor from each group group. Factor common factor factor from each group. 2x  3 Now the common factor is (2x  3). Underline it. + 8x 3  12x 2 + 14x  21 What to DoHow to Do It

14. Check Factors by FOIL What to DoHow to Do It  Check by multiplying multiplying by First Note DIFFERENT POWERS of O and I terms F 0 IL (4x 2 + 7)(2x  3) Outer Inner Last 8x 3  12x 2 + 14x  21 8x 3  21  12x 2 14x 8x 3  12x 2 + 14x  21

15. THE END 5.1