$(x^2-4)^2\cdot(x^2-6x-7)=0$
$S.S.=\{x_1,x_2,...,x_n\}$
$\Rightarrow n=?$
Answer is given as $4$. I think it should be $6$ because of multiplicity of the roots. I debated this problem with my classmates but we can't reach a verdict.
What is the number of elements in the Solution Set of this problem?
On
It should be 4, because you can expand the whole thing out, and get a sextic equation, and there are 2 answers that each is seen 2 times. The four answers are 2, -2, and the two different roots of $x^2-6x-7=0$. Or, just 6-2=4.
On
This polynomial can be simplified to:
$$(x-2)^2(x+2)^2(x+1)(x-7)$$
Hence it has repeated roots at $x=\pm 2$ and non-repeated roots at $x=1$ and $x=7$.
On
It comes down to how you're supposed to interpret the "S.S." notation.
I think the idea of writing the roots as members of a set was that you should represent the set using the minimal possible value of $n,$ hence $n = 4,$ because the solution set can be written $\{-2, -1, 2, 7\}.$
If you don't assume that $n$ is to be minimized, $n$ could be any integer greater than $3.$ For example, set $x_1 = -2,$ $x_2 = -1,$ $x_3 = 2,$ and $x_i = 7$ for $i = 4, 5, \ldots, 8.$ Then $n = 8$ and you can write the solution set as $\{-2, -1, 2, 7, 7, 7, 7, 7\},$ which is considered the same set as $\{-2, -1, 2, 7\}$ (your classmates' answer) or as $\{-2, -2, -1, 2, 2, 7\}$ (your answer).
This is one of the reasons I think some people ought to use words more often and mathematical symbols less. It's very simple to ask in words, "How many unique roots does this equation have?" The same question is much harder to write in symbolic form. The question as asked does not quite succeed at that. We have to make our best guess as to what meaning was intended, and it seems most people would guess $n =4$ (as I also would guess).
This seems to be a matter of semantics. When counted with multiplicity, the equation has $6$ roots. But the set of roots has only four elements, because there are only four distinct roots. That is to say, $$\{-2,-2,1,2,2,7\}=\{-2,1,2,7\},$$ because by definition a set does not contain duplicate elements.