Solving x^3-x-1=0: A Step-by-Step Guide

[SELFMADE]

How to solve this? Please help!

 

[symbolipoint]

If you can divide the polynomial expression by a linear binomial, then you can determine what value of x of each binomial will make the binomial equal to zero. Try first divisors x+1, and x-1.

 

[HallsofIvy]

By the "rational root theorem", which says that if m/n (m and n integers) is a rational root of $a_nx^n+ a_{n-1}x^{n-1}+\cdot\cdot\cdot+ a_1x+ a_0$ then n must divide the "leading coefficient", $a_n$, and m must divide the "constant term", a_0. That tells us that the only possible rational roots for this equation are the intgers 1 and -1. Unfortunately, trying them $1^3- 1- 1= -1$ and $(-1)^3- (-1)-1= -1$. Since neither of those is equal neither 1 nor -1 is a root and so this equation has no rational roots.

But if x= 2, $2^3- 2- 1= 5$ so there is clearly a root between 1 and 2. The only thing I could suggest is Cardano's "cubic formula": http://www.sosmath.com/algebra/factor/fac11/fac11.html.

 

[Epic Jeff]

i have a similar problem

x^2 -x -20 < 0

this is what i have so far:

x^2 - x < 20

x^2 - x < 20
_______x to eliminate the power

x - 1 < 20
_______x


i'm pretty much stuck there, any help?
p.s. underscore is just to put the X where i want it

 

[rock.freak667]

What exactly are you to do with that inequality? Find the range of values for x?



x - 1 < 20
_______x

I think at one point you divided by x to get this, but you can't just divide by x in an inequality, as the sign of x can change the inequality.

But if you want to get $x^2-x-20$ to have x to a single power then try completing the square.

 

[HallsofIvy]

i have a similar problem

x^2 -x -20 < 0

this is what i have so far:

x^2 - x < 20

x^2 - x < 20
_______x to eliminate the power

x - 1 < 20
_______x


i'm pretty much stuck there, any help?
p.s. underscore is just to put the X where i want it

Please do not "hijack" other people's threads for a new problem. And, I can see no similarity, except that they both involve polynomials.

I recommend you factor $x^2- x- 20$ which can be done relatively easily. Use the fact that the product of two numbers is negative if and only if one is positive and the other negative.

 

[Epic Jeff]

My bad, thanks for the help though.