math question that I need help with

Question

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. (Enter your answers as a comma-separated list.)

f(s) = 2s³ − 9s² + 30s − 50

Answer 1

rational factors will be a factor of the constant term over a factor of the coefficient of the leading term

such as 5/2, 5 is an integer factor of 50, 2 is a factor of 2.

2s^3 -9s^2 +30s -50

if s=0 then f(s) =-50

2(2^3) -9(2^2) +30(2)-50 = 16-36+60-50= -20+10=-10

2(3^3)-9(3^2) +30(3)-50=54-81+90-50 =13

since f(s) changes sign as s goes from 2 to 3, there is a zero inbetween somewhere, maybe around 2.5

and maybe 3 zeros inbetween

by DeCartes' rule, there are a potential 3 positive real zeros, as the polynomial has 3 sign changes

and zero negative real zeros, because there are zero sign changes if you replace x with -x

that means either 1 positive real or 3 positive real zeros

if only 1 positive real, then also 2 imaginary zeros

graph the equation in Desmos or a graphing calculator and see where the curve intersects the x axis, those are the zeros

or plug the equation into an online equation calculator to solve

3rd degree equations are not easily solved

from graphing the equation, it looks like about 5/2 = 2 1/2 = 2.5 is the only zero

if you can

factor into 2 factors

(2x-5)(x^2-2x+10)

set each factor =0

x =5/2 = 2 1/2 is the real zero, 2 imaginary zeros found by solving x^2-2x+10

x =1 +/-(1/2)sqr(-36)

x= 1+3i, 1-3i

check the 5/2 as a zero

2(5/2)^3 -9(5/2)^2 + 30(5/2) -50

= 125/4 - 225/4 + 300/4 - 200/4

=(125-225+300-200)/4

= 0/4

=0

once you have one zero you also have one factor

x=5/2

2x=5

2x-5 =0

with 2x-5 as a factor, divide it into the cubic polynomial to get another factor

(2x^3-9x^2+30x-50)/(2x-5) = x^2 -2x +10. you can do that with synthetic or long division