Andy C. answered • 08/18/17

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since it is degree 3 equation, there must be 3 roots.

part 2:

Descartes rule of signs states the number of positive roots is

either equal to the number of sign changes or less than the

number of sign changes by an even number.

Here's the link:https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

The sign changes twice, so there are at most two positive roots

part 3: for negative roots, x is replaced by -x

The polynomial equation becomes -x^3 - 7x^2 +x + 7

Again the sign changes only once so there is at most one negative root.

Now we solve it:

x^3 - 7x^2 - x + 7 = 0

x^2 ( x - 7) - ( x - 7) = 0 <--- factors -1 out of the last two terms

(x-7) ( x^2 - 1) = 0 <--- factors out (x-7) by grouping

(x-7)(x + 1)(x-1) = 0

x-7 = 0 or x + 1 = 0 or x-1 = 0

x=7 or x= -1 or x = 1

A simple check by inspection and mental arithmetic

makes it easy to verify that these solutions are correct.

For example, x=1 ---> 1^3 - 7*1^2 - 1 + 7 = 1 - 7 - 1 + 7 = 0

The check of the other two solutions are left for you.

The solution set is {7, 1, -1}

which as expected is two positive and one negative solution.