I have two related questions.
I was solving some problems on the matter of (Let's call it "rule X"):
$a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b \space + ... + \space ab^{n - 2} + b^{n - 1})$
And at some point I encountered this equation, where $x$ has to be found:
$(1 + x + x^2 + x^3)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) = (1 + x + x^2 + x^3 + x^4 + x^5)^2$
Considering the the topic, I did it this way:
- Multiplying both sides by $(x - 1)^2$
- Following the rule X we can already simplify the equation as following: $(x^4 - 1)(x^8 - 1) = (x^6 - 1)^2$
- Simplifying it further: $x^{12} - x^8 - x^4 + 1 = x^{12} - 2x^6 + 1 \implies x^8 + x^4 = 2x^6 \implies x^4(x^4 + 1) = 2x^6 \implies x^4 + 1 = 2x^2 \implies x^4 - 2x^2 + 1 = 0 \implies (x^2 - 1)^2 = 0$
And the valid values for $x$ are only $±1$
But, if we take the initial equation before the step (1), you can easily see that the only valid values for $x$ are $0$ and $-1$. I couldn't figure out what's wrong, because my calculations seemed correct, until I understood that I was actually just adding a new valid value for $x$. I thought that mathematically I didn't do anything wrong, since there's a simple rule that states we can multiply/divide/add/subtract, whatever algebraic operations we want, both sides of an equation on the same number/equation. So I'm left with two questions:
- Can you perform any additional algebraic operations on both sides of an equation if it changes the final answer ? If no, how do I solve the equation?
- Weather you can or no, coming from the way I solved this problem, there must be now three valid values for $x$: $-1$, $0$, and $1$, but as you saw I got only two valid values. Where was I wrong in my calculations ?
