laurakate said:

3x^3 + 2x^2 - 7x + 2

Find a root of the given polynomial by guessing or by trial-and-error

I think guessing IS trial-and-error.

Did you try this, yet?

Otherwise, what is your specific question about this exercise?

Here is some basic information that you should learn.

If some number r is a root of a polynomial, then the expression x - r is a factor of the polynomial.

Once you know one factor of a polynomial, you can use division to find remaining factors.

EG:

x^3 - 37x - 84

r = -3 is a root of this polynomial because substituting -3 for x causes the polynomial to evaluate to zero. (That's the definition of a root.)

(-3)^3 - 37(-3) - 84 = 0

Well, if -3 is a root, then x - (-3) is a factor of the polynomial. In other words, the expression x + 3 evenly divides the polynomial.

Using polynomial long-division, we get the following.

(x^3 - 37x - 84)/(x + 3) = x^2 - 3x - 28

The two factors of this quadratic quotient can be found using your favorite method.

Again, please ask specific questions, so that we might determine where to begin helping you.

Cheers ~ Mark