Question

Reduce the polynomial and find its complex roots
\(f(x)=x^4+5x^3-9x^2-85x-136\)

Answer 1

lol didnt know u were on

Answer 2

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

Answer 3

By reduce do you mean factor?

Answer 4

i am assuming so

Answer 5

Well, I have a couple ideas, so let's see if I can do this by hand...

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

Answer 6

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.

Answer 7

Let me just see if I can follow what u did

Answer 8

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).

Answer 9

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

Answer 10

whoops cubed lol

Answer 11

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

Answer 12

Thanks :)

Answer 13

welcome

Answer 14

OKKKK so like will this method always work?

Answer 15

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.

Answer 16

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

Answer 17

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.

Answer 18

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