Question
The function has at least one integer root.
Use the integer root theorem to find that root, then proceed to find all
complex roots. (Note: roots may be integer, rational, irrational, and/or
complex.)
Answer Attempt 3 out of 10
There are three roots :

To find integer roots, we can use the Rational Root Theorem, which suggests that any rational solution must be a factor of the constant term divided by a factor of the leading coefficient. For the polynomial , the constant term is 2, and the leading coefficient is 3. The possible integer roots can be . Testing gives us , so is indeed a root!

Now, we can use polynomial long division or synthetic division to factor by . This leads us to . We can now find the remaining roots by setting . Using the quadratic formula:

Thus, we get two roots: and . The complete list of roots is .

These roots not only tell us the intersections along the x-axis, but they also serve as stepping stones in higher math. Understanding polynomial roots is foundational for fields such as algebra, calculus, and even engineering! Roots can be used in signal processing, optimization problems, and even economic modeling to analyze trends and forecast behaviors in data.

If you ever feel overwhelmed with finding roots, just remember to check small integers first and keep your calm! Mark common mistakes, such as neglecting to factor correctly or overlooking updates in signs during calculations. With patience and attention to detail, you will reach the solution like a math wizard!