Consider the polynomial f x x3 2x2 2x 3 List all the pos

Consider the polynomial f (x) = x^3 - 2x^2 - 2x - 3 List all the possible rational zeros. Find one rational zero Divide by the factor corresponding, to part b to reduce the polynomial Factor the polynomial completely.

Solution

f(x)= x³-2x²-2x-3

a. To find the possible zeroes,we have to use rational root test

Here constant is 3 and leading coefficient is 1

Factors of constant = +-1,3

Factors of leading coefficient +-1

Therefore possible rational zeroes are +-1,3/1= +-1,3

b. If we plug x=3,we get

f(3)= 3³-2(3)²-2(3)-3= 27 -18 -6-3=0

Therefore one rational zero is x=3

c. To find the other zeroes we have to divide the given polynomial by x-3

And on doing that we get

(x²+ x + 1)

d. f(x)= (x-3)(x²+x + 1)