Algebra solve for $x$ in the equation $x^3 = x$

Question

I tried to solve for x in the equation $x^3=x$. I did

$$x^3=x$$

$$x^2=1$$

$$x=\pm1$$

but it's wrong, can anyone help.

You can't divide by $x$ if $x=0$. Hence, you must consider the possibility that $x=0$ independently. And it turns out that $x=0$ also works, so that should give you an answer. — Sarvesh Ravichandran Iyer, Sep 12 '16 at 23:47
You have two thirds of it but you forgot a very important and critical step. If $x = 0$ you can not do step two. So you must consider Case 1: x = 0 then x^3 = x = 0 is acceptable. Case 2: x \ne 0. then x = +/- 1. So x = 0,1,-1. THREE possible answers. — fleablood, Sep 12 '16 at 23:51

score 11 · Accepted Answer · edited Feb 25 '23 at 18:55

11

You forgot $0$. You shouldn't divide by $x$, or else you will miss it. You can solve it like this:

$$x^3=x$$ subtract $x$ from both sides$$x^3-x=0$$ factor out $x$ $$x(x^2-1)=0$$factor $x^2-1$ $$x(x+1)(x-1)=0$$ Simplify $$x=-1,0,1$$

edited Feb 25 '23 at 18:55

Sine of the Time

4,931

answered Sep 12 '16 at 23:47

suomynonA

6,895

3

That's nice. It avoids the case breaking divide by zero| x is 0 which is so hard to explain without confusing people. – fleablood Sep 12 '16 at 23:52
This is clear and helpful, I get it now. Thanks! – thunderbolt Sep 12 '16 at 23:54

Alex Ortiz · Answer 2 · 2016-09-13T00:02:51.613

1

The polynomial $p(x) = x^3 - x$ has at most three zeroes by the Fundamental Theorem of Algebra. By inspection, $1, -1, 0$ are three zeroes of $p$: $$ p(1) = 1^3 - 1 = 0 \\ p(-1) = (-1)^3 - (-1) = -1 + 1 = 0 \\ p(0) = 0^3 - 0 = 0, $$ and by the Fundamental Theorem of Algebra, these are the only zeroes of $p$.

edited Sep 13 '16 at 00:02

answered Sep 12 '16 at 23:47

Alex Ortiz

24,844

Do you want to say a bit more about how you find those zeros? – Thomas Sep 12 '16 at 23:59
1

I don't think it is evident for the OP. – Thomas Sep 13 '16 at 00:40
@Thomas Maybe you should consider that the other answer to this question already showed a method of computing the zeroes. The OP may also benefit from seeing someone point out that in certain cases, we don't need to do any hard work if we spend a few extra moments pondering the nature of the problem and other facts we already know. The point of downvoting someone's answer is to send the message that the answer is not useful. In light of others' answers, my answer would be truly useless if I just computed the zeroes like the other answer. – Alex Ortiz Sep 13 '16 at 00:52

Algebra solve for $x$ in the equation $x^3 = x$

2 Answers2