## swissgirl 3 years ago Reduce the polynomial and find its complex roots $$f(x)=x^4+5x^3-9x^2-85x-136$$

1. swissgirl

lol didnt know u were on

2. swissgirl

not quite sure how to reduce it since u start dealing with decimals. It shld be quite simple though

3. KingGeorge

By reduce do you mean factor?

4. swissgirl

i am assuming so

5. KingGeorge

Well, I have a couple ideas, so let's see if I can do this by hand... $f(x)=x^4+5x^3-9x^2-85x-136\\ f(x)=x^4-17x^2+5x^3+8x^2-17(5x-8)\\ f(x)=x^4-17x^2+x^2(5x+8)-17(5x+8)\\ f(x)=x^4-17x^2+(x^2-17)(5x+8)\\ f(x)=x^2(x^2-17)+(x^2-17)(5x+8)\\ f(x)=(x^2-17)(x^2+5x+8)$

6. KingGeorge

So you can use the quadratic formula on $$x^2+5x+8$$ to find the complex roots, and just $x^2-17=0\implies x=\pm\sqrt{17}$For the real roots.

7. swissgirl

Let me just see if I can follow what u did

8. KingGeorge

Take your time. I'm honestly very surprised I was able to factor that almost entirely by hand. (I used wolfram to check correctness, and to find the gcd of 85, 136).

9. swissgirl

idk if its possible. Dont drive urself crazy buttt I cant decipher what is squred and what is tripled. Is it possible to make it larger

10. swissgirl

whoops cubed lol

11. KingGeorge

$\Large f(x)=x^4+5x^3-9x^2-85x-136\\ \Large f(x)=x^4-17x^2+5x^3+8x^2-17(5x-8)\\ \Large f(x)=x^4-17x^2+x^2(5x+8)-17(5x+8)\\ \Large f(x)=x^4-17x^2+(x^2-17)(5x+8)\\ \Large f(x)=x^2(x^2-17)+(x^2-17)(5x+8)\\ \Large f(x)=(x^2-17)(x^2+5x+8)$

12. swissgirl

Thanks :)

13. KingGeorge

welcome

14. swissgirl

OKKKK so like will this method always work?

15. KingGeorge

Not always. This is just factoring by grouping, and while potentially useful, it is often difficult to identify the correct time to use it, and what to group together. I just got lucky that I guessed the right things to group. For example, if I had $\Large f(x)=x^4+5x^3-10x^2-85x-136$instead, I would not be able to group it as easily.

16. swissgirl

ohhhh i seee. I guesss I have to play around. Not as simple as I wish it wld be. Thanks KG:)))))))))))

17. KingGeorge

You're welcome. I will note however, that if they tell you to factor something, it's usually possible to do it by hand with a little cleverness.

18. swissgirl

Haha So I hope and pray that it will be so. lol Thanks KGGGG