Question

show that polynomial
3x^3 - 5x^2 - 5x - 1 has no intergal zero​

Answer 1

Answer:

So, you want me to show that there are no integer roots of the cubic equation:

3x3+8x2−1=0

From the integral zero theorem, any integer roots must be factors of -1. This means we have two possibilities: i) x=1 ; and ii) x=−1

Let’s evaluate your cubic expression at these values.

i): 3×13+8×12−1=3+8−1=10≠0

ii): 3×−13+8×−12−1=−3+8−1=4≠0

Thus, there are no integer roots.

A little bit of trial and error, and you should be able to find oou that one root is:

x=13

We can thus factorise the cubic as (3x−1)(x2+3x+1)

Using the formula for finding the roots of the quadratic term, we have:

x=−32±5√2

So, we have three distinct roots, none of which are integers.