real zeros of the polynomial function

im studying for a test and I have this review sheet but im stuck on this problem and need help.

find all real zeros of the polynomial function. Determine the multiplicity of each zero: -2(x^2)+50(x^4)+3(x^3)-75(x^5)

i know some of it but i get stuck in the middle. Help please?

Hey,

Quote:

Originally Posted by hammerbox

im studying for a test and I have this review sheet but im stuck on this problem and need help.

find all real zeros of the polynomial function. Determine the multiplicity of each zero: -2(x^2)+50(x^4)+3(x^3)-75(x^5)

i know some of it but i get stuck in the middle. Help please?

$-2(x^2)+50(x^4)+3(x^3)-75(x^5)$

First:

$-2(x^2)+50(x^4)+3(x^3)-75(x^5)$

$= x^2 [-2+50x^2+3x - 75x^3]$

$= x^2 [-75x^3 + 50x^2 + 3x -2] = 0$

$<=>x^2 = 0 \vee [-75x^3 + 50x^2 + 3x -2] = 0$

I guess you do not know how to solve [-75x^3 + 50x^2 + 3x -2] = 0

Well, do you know Cardano's method or Newton's method to determine the zeros? If you don't I suggest you guess a zero, after that you use long polynomial division. So what methods do you know to find zeros?

Solutions are:
x1 = - 1/5
x2 = 1/5
x3 = 2/3
x4,5 = 0

Regards, Rapha

I haven't learned any of those methods nor long polynomial division in my class. Is there any other way to get the zeros other than those 3 ways?