CHAPTER 8.3 Objective One

Factoring Polynomials in the form of ax2+bx+c using trial factors.

The coefficient of x2 is not 1.

Therefore, factors of the coefficient of the x2 and the last term must be considered in factoring the trinomial. Hence, a factoring by trial and error method may have to be implemented.

The factoring procedures previously used in Chapter 8 will also apply to factoring trinomials in the form of ax2+bx+c.

Note: if the trinomial does not have a common factor then the binomials cannot have a common factor. Also, if both first and both last terms of the binomials are even then the middle term of the trinomial cannot be odd.

Factor 3x2+20x+12 1st set up binomials with the signs. ( + )( + ) 2nd insert factors of x2 ( x + )( x + ) Factors of 3 12 1,3 1,12 2,6 3,4 Use trial and error (3x+1)(x+12) FOIL = 3x2+37x+12 (3x+12)(x+1) = 3x2+15x+12 (3x+2)(x+6) = 3x2+20x+12 (3x+6)(x+2) = 3x2+12x+12 (3x+3) (x+4) = 3x2+15x+12 (3x+4)(x+3) = 3x2+13x+12 Note: binomial has common factors, so do not have to be considered.

Factor 6x2+11x+5

1st set up binomials with the signs. ( + )( + ) 2nd insert factors of x2 ( x + )( x + ) Factors of 6 5 1,6 1,5 2,3 Use trial and error (6x+1)(x+5) FOIL = 6x2+31x+5 (x+1)(6x+5) = 6x2+11x+5 (3x+1)(2x+5) = 6x2+16x+5 (2x+1)(3x+5) = 6x2+13x+5

Factor 6x2-5x-6 1st set up binomials with the signs. ( + )( - ) 2nd insert factors of x2 ( x + )( x - ) Factors of 6 - 6 1,6 -1,6 2,3 1,-6 -2,3 2,-3 Use trial and error (6x-1)(x+6) FOIL = 6x2+35x -6 (6x+1)(x- 6) = 6x2- 35x -6 (6x+2)(x-3) = 6x2-16x -6 (6x-2)(x+3) = 6x2+16x -6 (3x-2) (2x+3) = 6x2+5x -6 (3x+2)(2x -3) = 6x2-5x -6

Factor 8x2+14x-15 1st set up binomials with the signs. ( + )( - ) 2nd insert factors of x2 ( x + )( x - ) Factors of 8 -15 1,8 -1,15 2,4 1,-15 -3,5 3,-5 Use trial and error (8x+1)(x-15) FOIL = 8x2-119x-15 (8x-1)(x+15) = 8x2+119x-15 (8x-3)(x+5) = 8x2+37x-15 (8x+3)(x-5) = 8x2-37x-15 (4x+3) (2x-5) = 8x2-14x-15 (4x-3)(2x+5) = 8x2+14x-15

Factor 15-2x-x2

1st set up binomials with the signs. ( + )( - ) 2nd insert factors of x2 ( + x )( - x) Factors of 15 -1 1,15 1,-1 3,5 Use trial and error (1+x)(15 - x) FOIL = 15+14-x2

(15+x)(1- x) = 15-14x-x2

(3+x)(5 - x) = 15+2x-x2 (3 -x)(5+ x) = 15 -2x-x2

Factor 24-2x-x2

1st set up binomials with the signs. ( + )( - ) 2nd insert factors of x2 ( + x )( - x) Factors of 24 -1 1,24 1,-1 2,12 3,8 4,6 Use trial and error (1+x)(24 - x) FOIL = 24+23x-x2

(2+x)(12- x) = 24+10x-x2

(3+x)(8 - x) = 24+5x-x2 (4 -x)(6+ x) = 24 -2x-x2

Factor 3x3-23x2+14x = x (3x2-23x+14)

1st set up binomials with the signs. x( - )( - ) 2nd insert factors of x2 x( x - )( x - ) Factors of 3 14 1,3 -1,-14 -2,-7 Use trial and error x(3x-1)(x-14) FOIL = x(3x2-41x+14) x(3x-14)(x-1) = x(3x2-17x+14) x(3x-2)(x-7) = x(3x2-23x+14 ) x(3x-7)(x-2) = x(3x2-13x+14 )

Factor 4y2x2-30y2x+14y2 =

2y2(2x2-15x+7) 1st set up binomials with the signs. 2y2 ( - )( - ) 2nd insert factors of x2 2y2 ( x - )( x - ) Factors of 2 7 1,2 -1,-7 Use trial and error 2y2 (2x-1)(x-7) = 2y2(2x2-15x+7) 2 y2 (2x-7)(x-1) = 2y2(2x2-9x+7)

NOW YOU TRY!

1. 5x2-2x-3 (5x+3)(x-1) 2. 3x2+x-10 (3x-5)(x+2) 3. -12x3 -18x2+30x -6x(2x+5)(x-1) 4. 6x2+13x+6 (3x+2)(2x+3)

CHAPTER 8.3 Objective 2

AT times factoring by trial and error can be time consuming.

There is an alternative method to factoring trinomials in the form of ax2+bx+c; where a,b are the coefficients of the x terms and c is generally a constant.

The method that will be discussed breaks the trinomial into four terms, and factoring by grouping will be used.

Recall: Factor 3y3-4y2-6y+8

Try grouping into binomials to find a binomial factor (sometimes monomials must be rearranged to get binomial factors).

GCF y2(3y3- 4y2) GCF -2(-6y+8) y2(3y- 4) -2(3y-4) Factor (3y-4)[y2(3y-4)-2(3y-4)] Divide by GCF (3y-4) (3y-4) (3y-4) [y2 -2]

(3y-4) (y2 -2)

When factoring ax2+bx+c by grouping.

1st Multiply coefficient of x2 and the constant.

2nd Consider the factors of (a)( c) that sum to the middle term. (like factoring x2+bx+c)

3rd Rewrite the middle term with the factors derived in step two.

4th Factor by grouping.

Factor 2x2+19x-10 by grouping method.

1st (a)(c) = (2)(-10) = -20 2nd Consider factors of -20 -1, 20 1,-20 -2, 10 2,-10 -4, 5 4, -5 3rd Rewrite middle terms 2x2+20x – x -10 4th Factor by grouping 2x (x+10) -1(x+10) (x+10)(2x-1) Check by F.O.I.L. 2x2+19x-10

Factor 2x2+13x-7 by grouping method.

1st (a)(c) = (2)(-7) = -14 2nd Consider factors of -14 -1, 14 1,-14 -2, 7 2, -7 3rd Rewrite middle terms 2x2+14x – x -7 4th Factor by grouping 2x (x+7) -1(x+7) (x+7)(2x-1) Check by F.O.I.L. 2x2+13x -7

Factor 8x2-10x-3 by grouping method.

1st (a)(c) = (8)(-3) = -24 2nd Consider factors of -24 -1, 24 1,-24 -2, 12 2,-12 -3, 8 3, -8 -4, 6 4, -6 3rd Rewrite middle terms 8x2+2x –12x - 3 4th Factor by grouping 2x (4x+1) -3(4x+1) (4x+1)(2x-3) Check by F.O.I.L. 8x2-10x- 3

Factor 4x2-11x-3 by grouping method.

1st (a)(c) = (4)(-3) = -12 2nd Consider factors of -12 -1, 12 1,-12 -2, 6 2, -6 -3, 4 3, -4 3rd Rewrite middle terms 4x2+ x – 12x - 3 4th Factor by grouping x(4x+1) -3(4x+1) (4x+1)(x-3) Check by F.O.I.L. 4x2-11x-3

Factor 24x2y-76xy+40y by grouping method. Factor GCF = 4y(6x2-19x+10) 1st (a)(c) = (6)(10) = 60 2nd Consider factors of 60 -1,- 60 -2,- 30 -3, -20 -4, -15 -6, -10 3rd Rewrite middle terms 4y[6x2-4x –15x+10] 4th Factor by grouping 4y[2x (3x-2) -5(3x-2)] 4y(3x-2)(2x-5) Check by F.O.I.L. 4y(6x2-19x+10)

Factor 15x3+40x2-80x by grouping method. Factor GCF = 5x(3x2+8x-16) 1st (a)(c) = (3)(-16) = -48 2nd Consider factors of -48 -1, 48 1,-48 -2, 24 2,-24 -3, 16 3,-16 4,-12 -4, 12 6, -8 -6, 8 3rd Rewrite middle terms 5x [3x2+12x - 4x -16] 4th Factor by grouping 5x[3x (x+4) - 4(x+4)] 5x (x + 4)(3x - 4) Check by F.O.I.L. 5x (3x2+ 8x -16)

NOW YOU TRY!

1. 10x2+x - 2 (5x-2)(2x+1) 2. 12x2+31x +9 (3x+1)(4x+9) 3. 12x3y +10x2y -8xy 2xy(3x+4)(2x-1) 4. 25x2+41x+16 (Extra Credit) ???????????