Section 1.3 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 1*6 2*3 Choice to break up middle term 1x + 6x 2x + 3x What middle column simplifies to 7x 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 1*6 2*3 -1 * -6 -2 * - 3 Choice to break up middle term 1x + 6x 2x + 3x -1x + -6x -2x + - 3x I need to rewrite the -5x as -2x + (-3x) = x2 – 2x + (-3x) + 6 =(x2 – 2x) + (-3x + 6) = x(x-2) + (-3)(x-2) What middle column simplifies to 7x 5x -7x -5x Chapter 1: Factoring and Quadratic Equations = x(x-2) – 3(x-2) Solution: (x-2)(x-3) 5) y2 + 5y + 4 7) y2 – 5y + 4 9) z2 + 13z + 36 11) z2 – 13z + 36 13) x2 + 5x – 6 Factors of last number 1 * -6 -1*6 2*-3 -2*3 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) Choice to break up middle term 1x + -6x -1x + 6x 2x + -3x -2x + 3x What middle column simplifies to -5x 5x -1x 1x Chapter 1: Factoring and Quadratic Equations 15) x2 - 5x – 6 Factors of last number 1 * -6 -1*6 2*-3 -2*3 Choice to break up middle term 1x + -6x -1x + 6x 2x + -3x -2x + 3x What middle column simplifies to -5x 5x -1x 1x Choice to break up middle term 1x + -12x -1x + 12x 2x + -6x -2x + 6x 3x + -4x -3x + 4x What middle column simplifies to -11x 11x -4x 4x -1x 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) 17) x2 + 4x – 12 19) x2 – 4x –12 Factors of last number 1 * -12 -1*12 2*-6 -2*6 3*-4 -3*4 I need to pick the 2x + - 6x = x2 + 2x + (-6x) – 12 Chapter 1: Factoring and Quadratic Equations = (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 Factors of last number 1*2 Choice to break up middle term 1x + 2x What middle column simplifies to 3x There is no way to get a 7x. This tells me that this can’t factor. Solution: Prime 23) x2 – 7x + 2 Factors of last number -1 *- 2 Choice to break up middle term -1x + -2x What middle column simplifies to -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 1 * -11 -1* 11 Choice to break up middle term 1y + (-11y) -1y + 11y There is no way to get a 3y. This tells me that this can’t factor. Solution: Prime What middle column simplifies to -10y 10y Chapter 1: Factoring and Quadratic Equations 27) y2 – 3y – 11 Factors of last number 1 * -11 -1* 11 Choice to break up middle term 1y + (-11y) -1y + 11y There is no way to get a (-3y_. This tells me that this can’t factor. Solution: Prime #29-58: Factor each trinomial, state if a polynomial is prime. 29) x2 + 11x + 18 Factors of last number 1 * 18 2*9 3*6 What the add to 1 + 18 = 19 2 + 9 = 11 3+6=9 The lasts must be +2 and + 9 Solution: (x+2)(x+9) 31) c2 + 12c + 20 Factors of last number 1 * 20 2 * 10 4*5 Solution: (x+2)(x+10) 33) r2 + 6r + 8 What the add to 1 + 20 = 21 2 + 10 = 12 4+5=9 What middle column simplifies to -10y 10y Chapter 1: Factoring and Quadratic Equations 35) y2 – 10y +16 Factors of last number -1 * - 16 -2 * - 8 -4 * - 4 What the add to -1 + -16 = -17 -2 + -8 = -10 -4 + -4 = -8 Solution: (x-2)(x-8) 37) x2 – 9x + 20 Factors of last number -1 * - 20 -2 * - 10 -4 * - 5 What the add to -1 + -20 = -21 -2 + -10 = -12 -4 + -5 = -9 Solution: (x – 2)(x – 10) 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 -1 * - 3 What the add to -1 + -3 = -4 No factors give me the -2 that I need, so this does not factor. Solution: Prime 41) b2 + 4b – 5 Chapter 1: Factoring and Quadratic Equations 43) z2 + 5z – 6 Factors of last number 1*-6 -1 * 6 2* - 3 -2*3 What the add to 1 + -6 = -5 -1 + 6 = 5 2 + - 3 = -1 -2 + 3 = 1 Solution: (z – 1)(z+6) 45) x2 + 4x – 12 Factors of last number 1 * - 12 -1 * 12 2* -6 -2*6 3*-4 -3*4 What the add to 1 + -12 = -11 -1 + 12 = 11 2 + -6 = -4 -2 + 6 = 4 3 + -4 = -1 -3 + 4 = 1 Solution: (x-2)(x+6) 47) x2 – 2x – 15 Factors of last number 1 * -15 -1 * 15 3* - 5 -3*5 Solution: (x - 3)(x + 5) 49) a2 – 9a – 22 51) x2 – 6x – 16 What the add to 1 + -15 = -14 -1 + 15 = 14 3 + -5 = -2 -3 + 5 = -2 Chapter 1: Factoring and Quadratic Equations 53) x2 + 2x + 8 55) y2 – 2y + 5 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 -1 * - 2 What the add to -1 + -2= -3 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 #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) Chapter 1: Factoring and Quadratic Equations 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) 63) 3x2 +12x – 36 First factor out the GCF of 3. = 3(x2 + 4x – 12) Factors of last number 1 * - 12 -1 * 12 2* -6 -2*6 3*-4 -3*4 Solution: 3(x-2)(x+6) 65) 6z2 – 30z + 24 67) x3 + 6x2 – 7x First factor out the GCF of x. What the add to 1 + -12 = -11 -1 + 12 = 11 2 + -6 = -4 -2 + 6 = 4 3 + -4 = -1 -3 + 4 = 1 Chapter 1: Factoring and Quadratic Equations = x(x2 + 6x - 7) Factors of last number 1 * -7 -1 * 7 Solution: x(x-1)(x+7) What the add to 1 + -7= -6 -1 + 7 = 6 69) -2x3 – 10x2 + 12x 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 1*9 3*3 Solution: 2(y+1)(y+9) 73) 30z + 3z2 + 45 What the add to 1 + 9 = 10 3+3=6