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CHAPTER 8.3 Objective One

CHAPTER 8.3 Objective One. Factoring Polynomials in the form of ax 2 +bx+c using trial factors. The coefficient of x 2 is not 1.

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CHAPTER 8.3 Objective One

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  1. CHAPTER 8.3 Objective One Factoring Polynomials in the form of ax2+bx+c using trial factors.

  2. 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.

  3. 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.

  4. 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

  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

  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

  7. 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

  8. 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

  9. 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 )

  10. 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)

  11. 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)

  12. 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.

  13. Recall: Factor 3y3-4y2-6y+8 • Try grouping into binomials to find a binomial factor (sometimes monomials must be rearranged to get binomial factors). • GCFy2(3y3- 4y2) GCF-2(-6y+8) • y2(3y- 4) -2(3y-4) • Factor (3y-4)[y2(3y-4)-2(3y-4)] • Divide byGCF(3y-4) (3y-4) • (3y-4) [y2 -2] • (3y-4) (y2 -2)

  14. 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.

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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)

  20. 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)

  21. 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) • ???????????

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