Can somebody explain the rationale here, i saw the solution but couldn’t understand a bit about the about the possible value of x. Please help.
Let’s try an example. Suppose n = 1. What would be the answer? What if n = 2?
for the given case at n =1,x = k , only one solution. for even cases x^2 = k, or even x^n = k, there would be only one solution in every case which is x^1/n. How come for n=3 it has 2 solution?
Let’s look at the n = 2 case again. That would be something of the form
ax^4 + bx^2 + c = 0
This can have up to four solutions. Can you find an example? After all, x^2 = k can have up to two solutions (x = \pm \sqrt{k}).
Oh! i was under the impression the negative roots are out of scope of GRE, ETS account for +ve roots, only. That’s why in in last line of the solution when it says (verabtim): “in any other cases (for example, when n=2), there can be only one value of x” (see solution image), I interpreted it as for even x^n = k, there would be only one solution in every case which is x^1/n. even for x^2 = k, x = root k, ONLY.
What really spooked me was for n=3, it has 2 solution for other case it has one.
Can you please explain if possible in detail with hypothetical numbers or equation?
also, i do get your point, for 4 degreed polynomial it can have four solution (and can touch x axis 4 times), and extrapolating this logic possible value of roots = degree of solution, for n = 5, equation will have degree 10 and hence possible 10 roots which can touch x axis and hence greater than 5. I dont get it why not? why 4 is the max one even degreed equation can touch the x axis?
Requesting for detailed and first principle based explanation.
We’ve updated the solution to make it clearer - can you take another look at it and see if that helps?
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Thanks a ton, now the solution became more comprehensible to be. Just to reiterate my understanding with hypothetical example, to confirm:
if the polynomial is ax^10 + bx^5+ c , then the max no of roots would be 2 and if the polynomial is ax^20+ bx^10+ c, then the max no of roots would be 4. (using the suggested logic)
Correct.