## jaersyn 2 years ago find all solutions of the polynomial equation: x^4 - x^3 - 6x^2 + 4x + 8 = 0

1. jenn0814

2. cammyabbo

you want to use grouping to factor this polynomial

3. cammyabbo

for the first 3 terms, you can factor out an X^2

4. jaersyn

how should i group them?

5. jaersyn

i got x^2(x^2-x-6)+4(x+2)

6. jaersyn

x^2(x-3)(x+2)+4(x+2) (x+2)(x^2(x-3)+4) :/

7. cammyabbo

lets start over

8. cammyabbo

x^4 - x^3 - 6x^2 + 4x + 8 = 0

9. matricked

use factor and remainder theorem to solve it

10. cammyabbo

x^2(x^2 - x - 6) + 4(x + 2) = 0

11. jaersyn

do you have to use special factors?

12. matricked

see by plugging x=-2 the expression x^4 - x^3 - 6x^2 + 4x + 8 becomes 0 hence x+2 is one of the factor for getting other factors divide x^4 - x^3 - 6x^2 + 4x + 8 by x+2

13. cammyabbo

no... (x^2+4)(x-3)(x+2)^2=0 now solve each equal to zero

14. cammyabbo

no to special factors, matricked is right also

15. jaersyn

how'd you get there..

16. cammyabbo

x=-2 and x=2 x=3

17. jaersyn

you skipped a couple steps from x^2(x^2 - x - 6) + 4(x + 2) = 0

18. cammyabbo

i further factored (x^2-x-6) and grouped (x^2+4)

19. jaersyn

can you show me how, from there, you got (x^2+4)(x-3)(x+2)^2=0?

20. cammyabbo

now set each equation equal to 0

21. cammyabbo

the x^2 and +4 you pulled out go in their own equation

22. jaersyn

wait so from x^2(x-3)(x+2) + 4(x+2) = (x^2+4)(x+2)^2 (x-3)?

23. cammyabbo

yes

24. cammyabbo

now set each equal to zero by themselves and solve for x

25. jaersyn

hold on

26. cammyabbo

x^2+4=0

27. jaersyn

the rest i know how, but h\where's the logic in that move, where you can take x^2+4

28. cammyabbo

you factored it out by grouping

29. jaersyn

it is my understanding that in order to factor by grouping both pieces need to have common factors like, "4(x+2)" doesn't have a (x-3)

30. cammyabbo

are you learning imaginary numbers?

31. cammyabbo

right, but x^2-x-6 did

32. cammyabbo

you further factored

33. matricked

x^4 - x^3 - 6x^2 + 4x + 8=0 x^4 +2x^3 -3x^3 - 6x^2 + 4x + 8=0 X^3(x+2) -3x^2(x+2) +4(x+2)=0 (x+2)(x^3-3x^2+4)=0 (x+2)(x^3-2x^2-x^2+4)=0 (x+2)(x^2(x-2) - (x+2)(x-2))=0 (x+2)(x-2)(x^2-x-2)=0 (x+2)(x-2)(x^2-2x+x-2)=0 (x+2)(x-2)(x(x-2)+1(x-2))=0 or(x+2)(x-2)(x-2)(x+1)=0 now set each equal to 0 thus the roots are -1,-2,2,2

34. cammyabbo

wow, nice job