1 / 21

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.

bela
Download Presentation

CHAPTER 8.3 Objective One

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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) • ???????????

More Related