1. MTH 10905 Algebra Special Factoring Formulas and a General Review of Factoring Chapter 5 Section 5

2. Special Factoring  Describe the following. a 2 – b 2 a 3 + b 3 a 3 – b 3

3. Special Factoring  Difference in Two Squares a 2 – b 2 = (a + b)(a – b)  Sum of Two Cubes a 3 + b 3 = (a + b)(a 2 – ab + b 2 )  Difference of Two Cubes a 3 – b 3 = (a – b)(a 2 + ab + b 2 )

4. Difference in Two Squares a 2 – b 2 = (a + b)(a – b) Example: x 2 – 100 (x) 2 – (10) 2 a = x and b = 10 (x + 10)(x – 10) You can always check with the FOIL method Example: 64x 2 – 9 (8x) 2 – (3) 2 a = 8x and b = 3 (8x + 3)(8x – 3)

5. Difference in Two Squares a 2 – b 2 = (a + b)(a – b) Example: 49x 2 – 81y 2 (7x) 2 – (9y) 2 a = 7x and b = 9y (7x + 9y)(7x – 9y) You can always check with the FOIL method Example: 169x 4 – 9y 4 (13x 2 ) 2 – (3y 2 ) 2 a = 13x 2 and b = 3y 2 (13x 2 + 3y 2 )(13x 2 – 3y 2 )

6. Difference in Two Squares a 2 – b 2 = (a + b)(a – b) Example: x 6 – y 8 (x 3 ) 2 – (y 4 ) 2 a = x 3 and b = y 4 (x 3 + y 4 )(x 3 – y 4 ) You can always check with the FOIL method Example: 81x 2 – 9y 2 GCF = 9 9(9x 2 – y 2 ) 9[(3x) 2 – (y) 2 ] a = 3x and b = y 9(3x 2 + y)(3x – y)

7. Difference in Two Squares a 2 – b 2 = (a + b)(a – b) Example: p 4 – 1 (p 2 ) 2 – (1) 2 a = p 2 and b = 1 (p 2 + 1) (p 2 – 1) (p 2 + 1)(p + 1)(p – 1) You can always check with the FOIL method Also a difference in two squares (p) 2 – (1) 2

8. Sum of Two Cubes a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) Example: x 3 + 125 (x) 3 + (5) 3 a = x and b = 5 (x + 5) (x 2 – 5x + 5 2 ) (x + 5)(x 2 – 5x + 25) You can always check with the FOIL method Example: 125x 3 + 27y 3 (5x) 3 + (3y) 3 a = 5x and b = 3y (5x + 3y) ((5x) 2 – (3y)(5x) + (3y) 2 ) (5x + 3y)(25x 2 – 15xy + 9y 2 )

9. Difference of Two Cubes a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Example: z 3 – 216 (z) 3 – (6) 3 a = z and b = 6 (z – 6) (z 2 + 6z + 6 2 ) (z – 6)(z 2 + 6z + 36) You can always check with the FOIL method Example: 27m 3 – n 3 (3m) 3 – (n) 3 a = 3m and b = n (3m – n) [(3m) 2 + (3m)(n) + n 2 ] (3m – n)(9m 2 + 3mn + n 2 )

10. 1.If all of the terms of the polynomial have a GCF other than 1, factor it out. 2.If the polynomial has two terms determine if it is a difference of two squares or a sum or difference of two cubes. Factor using formula. 3.If the polynomial has three terms, factor using any method we have discussed. 4.If the polynomial has more than three terms, try factoring by grouping. 5.Examine your factored polynomial to determine whether the terms in any factors have a common factor. If yes, factor out any common factors. General Procedure for Factoring a Polynomial

11. Example: 5x 4 – 405x 2 GCF = 5x 2 5x 2 (x 2 – 81) 5x 2 (x + 9)(x – 9) You can always check with the FOIL method Example: 3p 2 q 2 + 12p 2 q – 36p 2 GCF = 3p 2 3p 2 (q 2 + 4q – 12) (3p 2 )(q – 2)(q + 6) General Procedure for Factoring a Polynomial Difference in Two Squares (x) 2 – (9) 2 Factor again Factors -12 add 4 (-2)(6) -2 + 6

12. Example: 3t 3 r 2 – 12tr 2 + 15r 2 GCF = 3r 2 3r 2 (t 3 – 4t + 5) You can always check with the FOIL method Example: 5rt + 5r – 15t -15 GCF = 5 5(rt + r – 3t – 3) 5 (r)(t + 1) – 3(t + 1) 5(t + 1)(r – 3) General Procedure for Factoring a Polynomial PRIME More than 3 terms so we can try Factor by Grouping at this point. Factors 5 add -4 (1)(5) 1 + 5 = 6 (-1)(-5) -1 + -5 = -6

13. Example: 45x 2 – 45x – 50 GCF = 5 5(9x 2 – 9x – 10) 5(9x 2 + 6x – 15x – 10) 5 [(3x)(3x + 2) – (5)(3x + 2)] 5(3x + 2)(3x – 5) You can always check with the FOIL method Example: 3m 4 n + 192mn GCF = 3mn 3mn(m 3 + 64) 3mn(m 3 + 4 3 ) 3mn(m + 4)(m 2 – 4m +16) General Procedure for Factoring a Polynomial Factor by Grouping a = 9 b = -9 c = -10 a c = 9 -10 = -90 Factor -90 add -9 (6)(-15) 6 + -15 Sum in Two Cubes (a 3 + b 3 ) (a + b)(a – ab +b 2 ) a = m b = 4

14. Remember  Learning to recognize special forms will help when you encounter them later.  There is no special form for the sum of two squares, only the difference of two squares has a special form.  When there are only two terms, you should try special forms for squares or cubes.  Try factor by grouping when you have more than three terms.  Sometimes trial and error is necessary in factoring, therefore the more you practice the more you will understand when to use trial and error and when to use another method.

15. HOMEWORK 5.5 Page 330: #13, 19, 25, 29, 33, 57, 58, 69, 71