Comparing two linear equation

Question

If

$$a+b+c = 0$$

then what is the value of

$$a^5 + b^5 + c^5$$

What mathematical concepts should I be good at in order to understand and solved problems like this?

Answer 1

\begin{align}a^5+b^5+c^5&=a^5+b^5-(a+b)^5\\ &=-5a^4b-5ab^4-10a^3b^2-10a^2b^3\\ &=-5ab(a^3+2a^2b+2ab^2+b^3)\\ &=-5ab(a+b)(a^2+ab+b^2)\\ &=5abc(a^2+ab+b^2)\end{align}

Now remember please, what did you want to solve?

Answer 2

Let $x$ be any real we can make $a+b+c=0$ and $a^5+b^5+c^5=x$.

For instance let set $a=\left(\frac{16}{15}x\right)^{\frac 15}$ and $b=c=-\frac 12a$

So there is absolutely no correlation between these two sums.

Now regarding vocabulary, $a+b+c$ is linear in $a,b,c$ but $a^5+b^5+c^5$ is not. $a+b+c=0$ is an equation because there is an equal sign so you constrain the variables in this sum to have certain values, but $a^5+b^5+c^5$ is just some number, not an equation.