Factoring Trinomials of the Form x2 + bx + c

Chapter 1: Factoring and Quadratic Equations Section 1.3: Factoring Trinomials of the Form x2 + bx + c #1 – 28: Rewrite as a polynomial with 4 terms (if possible) then factor by grouping and check your answer, state if a polynomial is prime. 1) x2 + 5x + 6 I need to break the 5x up into two terms that will make factoring by grouping work. I do this by first listing the factors of the 6. Factors of last number Choice to break up middle term What middle column simplifies to 1*6 1x + 6x 7x 2*3 2x + 3x 5x I will rewrite the problem using 2x + 3x, of course I could write 3x + 2x and still get the correct answer. = x2 + 2x + 3x + 6 Now think of this as (x2 + 2x) + (3x + 6) and factor by grouping. = x(x+2) + 3(x+2) Solution: (x+2)(x+3) 3) x2 – 5x + 6 Factors of last number Choice to break up middle term What middle column simplifies to 1*6 1x + 6x 7x 2*3 2x + 3x 5x -1 * -6 -1x + -6x -7x -2 * - 3 -2x + - 3x -5x 3 Chapter 1: Factoring and Quadratic Equations I need to rewrite the -5x as -2x + (-3x) = x2 – 2x + (-3x) + 6 ( I could have written x2 + (-2x) + (-3x) + 6) =(x2 – 2x) + (-3x + 6) (if I did I could write (x2 + (-2x)) + (-3x + 6) = x(x-2) + (-3)(x-2) = x(x-2) – 3(x-2) Solution: (x-2)(x-3) 5) y2 + 5y + 4 I need to break the 5y up into two terms that will make factoring by grouping work. I do this by first listing the factors of the 4. Factors of last number Choice to break up middle term What middle column simplifies to 1*4 1y + 4y 5y 2*2 2y + 2y 4y I will rewrite the problem using 1y + 4y, of course I could write 4y + 1y and still get the correct answer. = y2 + 1y + 4y + 4 Now think of this as (y2 + 1y) + (4y + 4) and factor by grouping. = y(y + 1) + 4(y + 1) Solution: (y + 1)(y + 4) 4 Chapter 1: Factoring and Quadratic Equations 7) y2 – 5y + 4 I need to break the -5y up into two terms that will make factoring by grouping work. I do this by first listing the factors of the 4. (I really only need the factors with negatives.) Factors of last number Choice to break up middle term What middle column simplifies to -1 *-4 -1y + -4y -5y -2 * -2 -2y + -2y -4y I will rewrite the problem using -1y +- 4y, of course I could write -4y + -1y and still get the correct answe. = y2 - 1y + (-4y )+ 4 Now think of this as (y2 - 1y) + ((-4y) - 4) and factor by grouping. = y(y – 1) + -4(y – 1) = y(y - 1) - 4(y - 1) Solution: (y – 4)(y – 1) 5 Chapter 1: Factoring and Quadratic Equations 9) z2 + 13z + 36 I need to break the 13z up into two terms that will make factoring by grouping work. I do this by first listing the factors of the 36. Factors of last number Choice to break up middle term What middle column simplifies to 1 * 36 1z + 36z 37z 2*18 2z + 18z 20z 3*12 3z + 12z 15z 4*9 4z + 9z 13z 6*6 6z+ 6z 12z I will rewrite the problem using 4z + 9z = z2 + 4z + 9z + 36 Now think of this as (z2 + 4z) + (9z + 36) and factor by grouping. = z(z + 4) + 9 (z + 4) Solution: (z + 9)(z + 4) 6 Chapter 1: Factoring and Quadratic Equations 11) z2 – 13z + 36 I need to break the -13z up into two terms that will make factoring by grouping work. I do this by first listing the factors of the 36. Factors of last number Choice to break up middle term What middle column simplifies to -1 * -36 -1z + -36z -37z -2*-18 -2z + -18z -20z -3*-12 -3z + -12z -15z -4*-9 -4z + -9z -13z -6*-6 -6z+ -6z -12z I will rewrite the problem using -4z + -9z = z2 + (-4z) + (-9z) + 36 Now think of this as (z2 + (-4z)) + ((-9z) + 36) and factor by grouping. =z(z + (-4)) + -9(z – 4) = z(z - 4) - 9(z -4) Solution: (z - 9)(z - 4) 7 Chapter 1: Factoring and Quadratic Equations 13) x2 + 5x – 6 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -6 1x + -6x -5x -1*6 -1x + 6x 5x 2*-3 2x + -3x -1x -2*3 -2x + 3x 1x I will pick the -1x + 6x = x2 + (-1x) + 6x – 6 = (x2 – 1x) + (6x – 6) = x(x-1) + 6(x-1) Solution: (x-1)(x+6) 8 Chapter 1: Factoring and Quadratic Equations 15) x2 - 5x – 6 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -6 1x + -6x -5x -1*6 -1x + 6x 5x 2*-3 2x + -3x -1x -2*3 -2x + 3x 1x I will pick the 1x + -6x = x2 + 1x + (-6x) – 6 = (x2 + 1x) + (-6x – 6) = x(x+1) +( -6)(x+1) =x(x+1) – 6(x+1) Solution: (x+1)(x-6) 9 Chapter 1: Factoring and Quadratic Equations 17) x2 + 4x – 12 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -12 1x + -12x -11x -1*12 -1x + 12x 11x 2*-6 2x + -6x -4x -2*6 -2x + 6x 4x 3*-4 3x + -4x -1x -3*4 -3x + 4x 1x I need to pick the -2x + 6x = x2 + (-2x) + 6x – 12 = (x2 + (-2x)) + (6x – 12) = x(x + (-2)) + 6(x – 2) = x(x-2) + 6(x-2) Solution: (x-2)(x + 6) 10 Chapter 1: Factoring and Quadratic Equations 19) x2 – 4x –12 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -12 1x + -12x -11x -1*12 -1x + 12x 11x 2*-6 2x + -6x -4x -2*6 -2x + 6x 4x 3*-4 3x + -4x -1x -3*4 -3x + 4x 1x Factors of last number Choice to break up middle term What middle column simplifies to 1*2 1x + 2x 3x I need to pick the 2x + - 6x = x2 + 2x + (-6x) – 12 = (x2 + 2x) + (-6x – 12) = x(x+2) + -6(x+2) = x(x+2) – 6(x+2) Solution: (x+2)(x-6) 21) x2 + 7x + 2 There is no way to get a 7x. This tells me that this can’t factor. Solution: Prime 11 Chapter 1: Factoring and Quadratic Equations 23) x2 – 7x + 2 ( I only need factors with negatives, ignore the double positive factors) Factors of last number Choice to break up middle term What middle column simplifies to -1 *- 2 -1x + -2x -3x There is no way to get a (-7x). This tells me that this can’t factor. Solution: Prime 25) y2 + 3y – 11 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -11 1y + (-11y) -10y -1* 11 -1y + 11y 10y There is no way to get a 3y. This tells me that this can’t factor. Solution: Prime 27) y2 – 3y – 11 Factors of last number Choice to break up middle term What middle column simplifies to 1 * -11 1y + (-11y) -10y -1* 11 -1y + 11y 10y There is no way to get a (-3y). This tells me that this can’t factor. Solution: Prime 12 Chapter 1: Factoring and Quadratic Equations #29-58: Factor each trinomial without grouping, state if a polynomial is prime. 29) x2 + 11x + 18 Factors of last number What the add to 1 * 18 1 + 18 = 19 2*9 2 + 9 = 11 3*6 3+6=9 The lasts must be +2 and + 9 Solution: (x+2)(x+9) 31) c2 + 12c + 20 Factors of last number What the add to 1 * 20 1 + 20 = 21 2 * 10 2 + 10 = 12 4*5 4+5=9 Solution: (c+2)(c+10) 33) r2 + 6r + 8 Factors of last number What the add to 1*8 1+8=9 2*4 2+4=6 The lasts must be +2 and + 4 Solution: (r+2)(r+4) 13 Chapter 1: Factoring and Quadratic Equations 35) y2 – 10y +16 Factors of last number What the add to -1 * - 16 -1 + -16 = -17 -2 * - 8 -2 + -8 = -10 -4 * - 4 -4 + -4 = -8 Solution: (x-2)(x-8) 37) x2 – 9x + 20 Factors of last number What the add to -1 * - 20 -1 + -20 = -21 -2 * - 10 -2 + -10 = -12 -4 * - 5 -4 + -5 = -9 Solution: (x – 4)(x – 5) 39) x2 – 2x + 3 I will consider the negative factors, as 1*3 has no chance of giving a negative. Also I don’t consider 1*-3 nor do I consider -1*3 as they multiply to -3 and do not multiply to positive 3. Factors of last number What the add to -1 * - 3 -1 + -3 = -4 No factors give me the -2 that I need, so this does not factor. Solution: Prime 14 Chapter 1: Factoring and Quadratic Equations 41) b2 + 4b – 5 Factors of last number What the add to 1 * -5 1 + -5 = -4 -1*5 -1 + 5 = 4 The lasts must be -1 and + 5 Solution: (b – 1)(b + 5) 43) z2 + 5z – 6 Factors of last number What the add to 1*-6 1 + -6 = -5 -1 * 6 -1 + 6 = 5 2* - 3 2 + - 3 = -1 -2*3 -2 + 3 = 1 Solution: (z – 1)(z+6) 15 Chapter 1: Factoring and Quadratic Equations 45) x2 + 4x – 12 Factors of last number What the add to 1 * - 12 1 + -12 = -11 -1 * 12 -1 + 12 = 11 2* -6 2 + -6 = -4 -2*6 -2 + 6 = 4 3*-4 3 + -4 = -1 -3*4 -3 + 4 = 1 Solution: (x-2)(x+6) 47) x2 – 2x – 15 Factors of last number What the add to 1 * -15 1 + -15 = -14 -1 * 15 -1 + 15 = 14 3* - 5 3 + -5 = -2 -3*5 -3 + 5 = 2 Solution: (x - 5)(x + 3) 16 Chapter 1: Factoring and Quadratic Equations 49) a2 – 9a – 22 Factors of last number What the add to 1 * -22 1 + -22 = -21 -1*22 -1 + 22 = 21 -2 * 11 -2 + 11 = 9 2 * -11 2 + -11 = -9 The lasts must be +2 and -11 Solution: (a + 2)(a – 11) 51) x2 – 6x – 16 Factors of last number What the add to 2 * -8 2 + -8 = -6 -2*8 -2 + 8 = 6 -4 * 4 -4 + 4 = 0 The lasts must be +2 and -8 Solution: (x + 2)(x – 8) 17 Chapter 1: Factoring and Quadratic Equations 53) x2 + 2x + 8 Factors of last number What the add to 1*8 1+8=9 2*4 2+4=6 No factors give me the 2 that I need, so this does not factor. Solution: prime 55) y2 – 2y + 5 I will consider the negative factors, as 1*5 has no chance of giving a negative. Also I don’t consider 1*-5 nor do I consider -1*5 as they multiply to -5 and do not multiply to positive 5. Factors of last number What the add to -1 * - 5 -1 + -5= -6 No factors give me the -2 that I need, so this does not factor. Solution: Prime 57) x2 – 5x – 9 Factors of last number What the add to 1 * -9 1 + -9 = -8 -1 * 9 -1 + 9 = 8 3* - 3 3 + -3 = 0 No factors give me the -5 that I need, so this does not factor. Solution: Prime 18 Chapter 1: Factoring and Quadratic Equations #59-74: Factor each trinomial. Make sure to factor out a negative or the GCF where applicable. 59) -x2 – 7x –10 First factor out a GCF of -1 = -1(x2 + 7x + 10) Factors of last number What the add to 1 * 10 1 + 10 = 11 2*5 2+5=7 Solution: =-1(x+2)(x+5) which also may be written -(x+2)(x+5) 61) -w2 + 18w – 77 First factor out a GCF of -1 = -1(w2 + 18w + 77) Factors of last number What the add to 1 * 77 1 + 77 = 78 11*7 11+7 = 18 Solution: =-1(w+11)(w+7) which also may be written -(w+11)(w+7) 19 Chapter 1: Factoring and Quadratic Equations 63) 3x2 +12x – 36 First factor out the GCF of 3. = 3(x2 + 4x – 12) Factors of last number What the add to 1 * - 12 1 + -12 = -11 -1 * 12 -1 + 12 = 11 2* -6 2 + -6 = -4 -2*6 -2 + 6 = 4 3*-4 3 + -4 = -1 -3*4 -3 + 4 = 1 Solution: 3(x-2)(x+6) 65) 6z2 – 30z + 24 First factor out the GCF of 3. = 6(z2 – 5z + 4) Factors of last number What the add to -1 * - 4 -1 + - 4 = - 5 -2 * - 2 -2 + - 2 = -4 Solution: 6(z – 1)(z – 4) 20 Chapter 1: Factoring and Quadratic Equations 67) x3 + 6x2 – 7x First factor out the GCF of x. = x(x2 + 6x - 7) Factors of last number What the add to 1 * -7 1 + -7= -6 -1 * 7 -1 + 7 = 6 Solution: x(x-1)(x+7) 69) -2x3 – 10x2 + 12x First factor out the GCF of -2x = -2x(x2 + 5x – 6) Now factor what is left inside the parenthesis. Factors of last number What the add to 1*-6 1 + -6 = -5 -1 * 6 -1 + 6 = 5 2* -3 2 + -3 = -1 -2*3 -2 + 3 = 1 The correct pair is the -1 + 6 Solution: -2x(x – 1)(x + 6) 21 Chapter 1: Factoring and Quadratic Equations 71) 20y + 18 + 2y2 First rewrite in descending power order. = 2y2 + 20y + 18 Next, factor out a GCF of 2. = 2(y2 + 10y + 9) Factors of last number What the add to 1*9 1 + 9 = 10 3*3 3+3=6 Solution: 2(y+1)(y+9) 73) 30z + 3z2 + 45 First rewrite in standard form = 3z2 + 30z + 45 Now factor out GCF of 3 3(z2 + 10z + 15) Now factor what’s left inside the parenthesis. Factors of last number What the add to 1 * 15 1 + 15 = 16 3*5 3+5=8 The inside of the parenthesis is prime. I can’t factor it more. The answer for the problem is not prime as I was able to factor out a 3. Solution: 3(z2 + 10z + 15) 22