1. MTH 10905 Algebra Factoring Trinomials of the form ax 2 + bx + c where a = 1 Chapter 5 Section 3

2. Factoring Trinomials  It is important that you understand sections 5.3 and 5.4 to be successful in Chapter 6.  In this section we learn factoring a trinomial in the form of ax 2 + bx + c where a = 1; for example, x 2 + 5x + 6  In section 5.4 we learn factoring a trinomial in the form of ax 2 + bx + c where a  1; for example, 2x 2 + 7x + 3

3. Trial and Error  When factoring x 2 + bx + c you will always get (x + ) (x + )  Write down the factors of the constant, c, and try them in the shaded area.  You need the factors of c that sum to b  Use the FOIL method to check

4. Trinomials in the form ax 2 + bx + c, a=1  It is important that you know what a, b and c equals.  Examples: x 2 + 7x + 12 = (x + 3)(x + 4)FOIL a = 1, b = 7, c = 12(3)(4) = 12 factors of 12 that sum to 73 + 4 = 7 x 2 – 2x – 24 = (x – 6)(x + 4) FOIL a = 1, b = -2, c = -24(-6)(4) = -24 factors of -24 that sum to -2(-6)+(4) = -2

5. Trial and Error Example: x 2 + 12x + 20 a=1 b=12 c=20 (x + 2)(x + 10) Factors of c: 20 Sum to b: 12 (1)(20) = 201 + 20 =21 (2)(10) = 202 + 10 = 12 (4)(5) = 204 + 5 = 9 You can always check using FOIL method (x + 2) (x + 10) (x)(x) + (x)(10)+ (2)(x) + (2)(10) x 2 + 10x + 2x + 20 x 2 + 12x + 20

6. Helpful Hint 1.If b = neg, c = pos, then factor = 2 negative 2.If b = neg, c = neg, then factor = 1 pos 1 neg 3.If b = pos, c = neg, then factor = 1 pos 1 neg 4.If b = pos, c = pos, then factor = 2 positive Example: Using x 2 + bx – c, determine the sign of the numbers in the factors: One positive and one negative factor

7. Factoring Trinomials Example: x 2 + x – 72 a=1 b=1 c=-72 (x + 9)(x – 8) Factors of c -72 Sum to b 1 (-1)(72)=-72-1 + 72 = 71 (-2)(36)=-72-2 + 36 = 34 (-3)(24)=-72-3 + 24 = 21 (-4)(18)=-72-4 + 18 = 14 (-6)(12)=-72-6 + 12 = 6 (-8)(9)=-72-8 + 9 = 1 Check using FOIL (x + 9) (x – 8) (x)(x) + (x)(-8)+ (9)(x) + (9)(-8) x 2 + (-8x) + 9x + (-72) x 2 + x – 72

8. Factoring Trinomials Example: x 2 – x – 72 a=1 b=-1 c=-72 (x – 9)(x + 8) Factors of c -72 Sum to b -1 (1)(-72)1 + -72 = -71 (2)(-36)2 + -36 = -34 (3)(-24)3 + -24 = -21 (4)(-18)4 + -18 = -14 (6)(-12)6 + -12 = -6 (8)(-9)8 + -9 = -1 Check using FOIL (x – 9)(x + (8) (x)(x) + (x)(8)+ (-9)(x) + (-9)(8) x 2 + 8x + (-9x) + (-72) x 2 – x – 72

9. Factoring Trinomials Example: x 2 – 11x + 30 a=1 b=-11 c=32 (x – 5)(x – 6) Factors of c 30 Sum to b -11 (-1)(-30)-1 + -30 = -31 (-2)(-15)-2 + -15 = -17 (-3)(-10)-3 + -10 = -13 (-5)(-6)-5 + -6 = -11 Check using FOIL (x – 5)(x – 6) (x)(x) + (x)(-6)+ (-5)(x) + (-5)(-6) x 2 + (-6x) + (-5x) + 30 x 2 – 11x + 30

10. Factoring Trinomials Example: t 2 + 4x – 32 a=1 b=4 c=-32 (t – 4)(t + 8) Factors of c -32 Sum to b 4 (-1)(32)-1 + 32 = 30 (-2)(16)-2 + 16 = 14 (-4)(8)-4 + 8 = 4 Check using FOIL (t – 4)(t + 8) (t)(t) + (t)(8)+ (-4)(t) + (-4)(8) t 2 + (8t) + (-4t) – 32 t 2 + 4t – 32

11. Factoring Trinomials Example: x 2 – 14x + 49 a=1 b=-14 c=49 (x – 7)(x – 7) (x – 7) 2 Factors of c 49 Sum to b -14 (-1)(-49)-1 + -49 = -50 (-7)(-7)-7 + -7 = -14 Check using FOIL (x – 7)(x – 7) (x)(x) + (x)(-7)+ (-7)(x) + (-7)(-7) x 2 + (-14x) + 49 x 2 – 14x + 49

12. Factoring Trinomials Example: x 2 – 2x – 63 a=1 b=-2 c=-63 (x – 9)(x + 7) Factors of c -63 Sum to b -2 (1)(-63)1 + -63 = -62 (3)(-21)3 + -21 = -18 (7)(-9)7 + -9 = -2 Check using FOIL (x – 9)(x + 7) (x)(x) + (x)(7)+ (-9)(x) + (-9)(7) x 2 + 7x + (-9x) – 63 x 2 – 2x – 63

13. Factoring Trinomials Example: x 2 + 10x + 20 a=1 b=10 c=20 PRIME Factors of c 20 Sum to b 10 (1)(20)1 + 20 = 21 (2)(10)2 + 10 = 12 (4)(5)4 + 5 = 9 A polynomial that cannot be factored using only integer coefficients is called a prime polynomial. No factors of c that can sum to b.

14. Factoring Trinomials Example: x 2 + 4xy + 4y 2 a=1 b=4 c=4 (x + 2y)(x + 2y) (x + 2y) 2 Factors of c 4 Sum to b 4 (1)(4)1 + 4 = 5 (2)(2)2 + 2 = 4 Check using FOIL (x + 2y)(x + 2y) (x)(x) + (x)(2y)+ (2y)(x) + (2y)(2y) x 2 + 2xy + 2xy + 4y 2 x 2 + 4xy + 4y 2 The last term of the factors must contain a y in order to get the y 2

15. Factoring Trinomials Example: x 2 – xy – 30y 2 a=1 b=1 c=-30 (x – 6y)(x + 5y) Factors of c -30 Sum to b -1 (1)(-30)1 + -30 = -29 (2)(-15)2 + -15 = -13 (3)(-10)3 + -10 = -7 (5)(-6)5 + -6 = -1 Check using FOIL (x – 6) (x + 5y) (x)(x) + (x)(5y)+ (-6y)(x) + (-6y)(5y) x 2 + 5xy + (-6xy) + (-30y 2 ) x 2 – xy – 30y 2 The last term of the factors must contain a y in order to get the y 2

16. Removing a Common Factor from a Trinomials Example: 3x 2 – 21x + 18 3(x 2 – 7x + 6) 3(x – 1)(x – 6) Factors of c 6 Sum to b -7 (-1)(-6)-1 + -6 = -7 Check using FOIL 3(x – 1)(x – 6) 3[(x)(x) + (x)(-6)+ (-1)(x) + (-1)(-6)] 3[x 2 + (-6x) + (-1x) + 6] 3[x 2 – 7x + 6] 3x 2 – 21x + 18 Sometimes each term has a GCF that we must pull out first making it easier to factor.

17. Removing a Common Factor from a Trinomials Example: 3m 3 + 9m 2 – 84m 3m(m 2 + 3m – 28) 3m(m + 7)(m – 4) Factors of c -28 Sum to b 3 (-1)(28)-1 + 28 = 27 (-2)(14)-2 + 14 = 12 (-4)(7)-4 + 7 = 3 Check using FOIL 3m(m + 7)(m – 4) 3m[(m)(m) + (m)(7)+ (7)(m) + (7)(-4)] 3m[m 2 + 7m + 7m + -28] 3m[m 2 + 14m - 28] 3m 3 + 9m 2 – 84

18. REMEMBER  Always factor out the GCF first.  A table can be helpful. Use one column for all possible factors of c an another column for the sum of the factors.  One or both factors of c can be negative.  When c is positive, the two factors will have the same sign as b.  When c is negative, the two factors will have opposite signs.  When the factors have opposite signs, the larger of the two will be the same sign as b  You should always check your work by multiplying.

19. HOMEWORK 5.3 Page 311: #17, 19, 21, 27, 39, 45, 63, 73