Question

Find all zeros of 2x^4 - 5x^3 +53x^2 - 125x + 75 =0

Answer 1

#1 DesCartes' Rule of Signs. Immediately, we have that there are NO negative Real zeros.

#2 Further, there MAY be 4, 2, or 0 Positive Real zeros. May as well start looking.

Factors of 75

1, 3, 5, 15, 25, 75

f(1) = 2 - 5 + 53 - 125 + 75 = 0 -- Whoops! We accidentally found one! There MUST be another!

Dividing f(x)/(x-1) gives g(x) = 2x^3 -3x^2 + 50x - 75

Searching again

g(1) = -26
g(3) = 102 -- That's interesting. There must be one between x = 1 and x = 3

Factors of 2

1, 2

We've done the '1' already.

1/2, 3/2 - Hold on! This is between 1 and 3!

g(3/2) = 0 -- We found another one!

From x = 3/2, we get 2x - 3 = 0.

g(x)/(2x-3) gives h(x) = x^2 + 50. Perfect. This is easily solved by the quadratic formula or anything else that can handle a quadratic equation.