Describe the zeros of the following function:

Question

f(x)=3x⁵-7x⁴+15x³-58x²-17x+6

^{Total zeros-}

^{Real zeros-}

^{Possible rational zeros-}

^{Positive real zeros-}

^{Negative real zeros-}

Answer 1

I think the answer is contained in Decartes' rule for polynomial zeroes.

As for first question, if the power of equation winding sown to a constant is n,

there are n possible vaues of x where the equatin is zerol (the roots)

decartes' rule,

look at equation, it changes sign four times,

that means there are either four real positive roots, or 4-2, or 4-4

that is number of positive roots equal 4, or 2, or zero

(the number of sign changes in equation or that number reduced by even integer).

now change all x to (-x) and substitute into equation and

count number of sign changes term to term. I count one. That

means there is definitely a negative real root since no even integer

subtracted from 1 provide answer greater than zero.