Factor this completely. 10 points to right answer! =]?

Question

x^4 - x^3 + x -1

factor it.

i need to know HOW to do it
if it helps anyone, the answer to it is (x-1)(x+1)(x^2 -x+1)

im desperate, lol i really need to know how to do this stupid problem.

10 points to best explanation!

Answer 1

I would try dividing it by (x - 1) by long division

Since the constant/last term in the expression is "1"
then the possible factors to try end in factors of 1,
or + 1 and - 1, so that is why you would try
dividing by (x + 1) or (x - 1) to find out which divides evenly:

x^4 - x^3 + 0x^2 + x - 1
-------------------------------
x + 1

= x^3 - 2x^2 + 2x - 1

so you have (x + 1)(x^3 - 2x^2 + 2x - 1)

then try dividing (x^3 - 2x^2 + 2x - 1)
by (x + 1) or (x - 1) using long division again

x^3 - 2x^2 + 2x -1
-----------------------
x - 1

= x^2 - x + 1

So the factors are
(x + 1)(x -1 )(x^2 - x +1)

Note: I didn't show the long division, but that is the trickiest part -- watching the minus signs when you multiply and subtract. It will take a while to type that, but I will try to add it later.

Answer 2

See this is not a stupid problem.
There is a rule how to solve it.

The rule is whenever you need to factorize such expressions who hv degree (highest power of x) greater than 2 then replace x by 0, 1 ,-1, 2, -2,3,-3 and so on untill you get zero as the value of the function.

Here if y=x^4 - x^3 + x -1
than at x=1
y=1-1+1-1=0
also at x=-1
y=-1+1-1+1=0
hence
(x-1)(x+1) are sureshot factors of x^4 - x^3 + x -1.
=> (x^2-1) are sureshot factors of x^4 - x^3 + x -1.
divide x^4 - x^3 + x -1 by x^2-1
you will get another factor.
That factor and (x^2-1) are factors of x^4 - x^3 + x -1.

Answer 3

1) Need to find x that can make e eqn = 0
In your case, is x=1 or -1. Therefore, we get these 2 eqns: (x-1) & (x+1)

2) Now, e eqn is this:
x^4 - x^3 + x -1 = (x-1)(x+1)(x^2 + kx + 1)
For e quadratic eqn,
x^4 = (x^2).(x).(x);
(-1) = (-1)(1)(1);
we need to find k by comparing coefficients -> LHS = RHS (whole eqn)
Eg: Comparing coeff. of x: 1 = -k + 1 - 1 -> k = -1
Therefore, e final ans is (x-1)(x+1)(x^2 - x + 1)

Answer 4

The arrows mean power x^4 means x to the power of 4 =)

Answer 5

I think it's X4-x3+x-1 or Xx4-Xx3-X1 (I think^ means take out what ever sign was there.But I'm ownly 12 so.).?

Answer 6

the arrows... are square roots. hmm, leme see.... nope not a clue. i dont do work over the summer.