How do I find the roots of x^5-5x+2 manually without using any calculators?

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How do I find the roots of x^5-5x+2 manually without using any calculators?

Mathematics
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Do you know the rational root theorem ?
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No I dont know the rational root theorem

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since this is a quintic, it's not a straight forward polynomial. you need to guess some of the roots to simplify this problem. to make a good guess you plug in the rational roots. the way you find it is this. a), find the factors of the constant term (1,2) b), find the factors of the leading term (1) all the combination of a/b is a possible root, so for this problem it is +/- 1 or +/-2. If they don't give you a root, try plugging in other easy numbers such as i or -i. If it still doesn't work, then this problem is beyond calculus level to do it by hand.
How do I know if these values +/- 1 and 2 give me a root? Do I plug it in back the equation?
that's it
and they must equal zero right?
nope. the rational root theorem states this... If the polynomial P(x) has the leading term with factors b and constant term with factor a, then a/b is a "possible" rational root, given that P(x) only has integer coefficients. so them failing is ok, it sucks for us though.
Is this the same theorem as the one where you divide the possible zeros by the equation using long division or synthetic division? So, if the reminder comes out to be zero than it that possible zero is in fact a zero?
What you are talking about is called the "factor theorem" and it is a very useful tool in these kind of problems. yes, plugging in the possible roots and getting 0 is the same thing as saying dividing by (x-c) where c is a possible root and having remainder equal to 0. rational root theorem tells you what possible roots are. factor theorem tells you that if (x-c) is a factor c is a root.

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