Question

List all possible rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely. Please show all of your work. f(x)=3x^4-28x^3+81x^2-84x+20

Answer 1

possible zeros are plenty
make fraction where numerator divides 20 and denominator divides 3

Answer 2

a pain because 20 has many divisors
\[\pm1,\pm2\pm4,\pm5,\pm10\pm20\] and then all of these divide by 3

Answer 3

actual zeros are much easier to find since this factors as
\[(x-5) (x-2)^2 (3 x-1)\]

Answer 4

so zeros are 2,5, 1/3

Answer 5

f(x)=3x^4-28x^3+81x^2-84x+20
f(-x)=3x^4+28x^3+81x^2+84x+20
\[\frac{p}{q}=\frac{\pm (1,2,4,5,10,20)}{\pm(1,3)}\]
Descartes rule of signs:
4 sign changes in f(x), so either 4,2 or 0 positive real roots
0 sign changes in f(-x), so no negative real roots
Don't both testing any of the negative values in the rational roots test.
After you find one root, use synthetic division and apply the roots test again, ignoring negatives