Question

Find all the zeroes of the equation. x4 – 6x2 – 7x – 6 = 0

Answer 1

Let's list the factors of the constant term, -6:
\[\Large\bf \pm 1,\;2,\;3,\;6\]And the factors of our leading coefficient, 1:\[\Large\bf \pm\color{#DD4747 }{1}\]

All of our rational roots will come from the ratio of these factors (Where the constant factors are in the numerator.
So our roots will come from these,\[\Large \bf \pm\frac{1}{\color{#DD4747 }{1}},\;\frac{2}{\color{#DD4747 }{1}},\;\frac{3}{\color{#DD4747 }{1}},\;\frac{6}{\color{#DD4747 }{1}}\]

Answer 2

This is a 4th degree polynomial, so 4 roots will exist.
Since we have a plus/minus sign in front of each number, we have 8 possibilities.

Answer 3

one positive and one negative value for each,
right?

Answer 4

Yes :)

Those fractions will simplify to :\[\Large \bf \pm1,\;2,\;3,\;6\]

Those are just `potential` roots though.
We have to do a little work to see which 4 of the 8 listed will actually solve the equation.

Answer 5

Since they're nice easy numbers, we should probably just plug them in one a time and see which ones work.\[\Large\bf x=1\qquad\qquad\to\qquad\qquad 1^4-6\cdot1^2-7\cdot1-6=0\]We want the problem to simplify to 0=0.
Does that happen with x=1?

Answer 6

hold on lemme check!

Answer 7

nope!

Answer 8

Hmmmm ok try \(\Large\bf x=-1\)
If that one doesn't work, try 2 and -2.
One of those numbers will work ^^ let's see if you can find it!

Answer 9

nopeeee ill try two, dang this is a lot of work hahah

Answer 10

Yah it's a 4th degree poly, so it's going to be a pain in the butt :c lol
3rd degree are hard enough >.<

Answer 11

-2 works :D

Answer 12

Ooo good!
So now that we've found our first `zero`, x=-2,
we can use that zero to find the others.

We don't want to continue checking the numbers listed because it's possible we could have repeated roots, in which case we wouldn't have any way of knowing D:

So from here, we can either employ `Polynomial Long Division` or `Synthetic Division`.
Are you familiar with either method?

Answer 13

i know of them but I'm not good at it/understand it really at all...

Answer 14

Let's try Synthetic, it's a little simpler to work with.