SOLUTION: According it Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots? i have no idea how to do this im trying to lea

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Question 92194This question is from textbook
: According it Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots? i have no idea how to do this im trying to learn this stuff on my own and its hard please help
f (a) = a^5 - 4a^2 - 7 (a)__________ (b)_________
This question is from textbook

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Using Descartes' Rule of Signs, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative)

First lets find the number of possible positive real roots:

For , simply count the sign changes

Here is the list of sign changes:

to (positive to negative)

(note: the rest of the terms have the same sign, so no extra sign changes occur)

So there is 1 sign change, this means there is a maximum of 1 positive root

So there is exactly one positive root

Now lets find the number of possible negative real roots

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First we need to find :

Plug in -a (just replace every "a" with "-a")

Simplify (note: if the exponent of the given term is odd, simply negate the sign of the term. If the term has an even exponent, then the sign of the term stays the same)

So

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Now lets count the sign changes for :
By looking at we can see that there are no sign changes (all the terms are negative).
So there are no negative roots

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Summary:

So there is 1 positive root and 0 negative roots