The zeroes of a polynomial are where the graph of the polynomial crosses the x-axis and why they are called x-intercepts. On graphs, you can also say these are points where \(y=0\), which means the same thing. These points have special meaning in many contexts.

One of the nice things about polynomials is that there are exactly the same number of roots as the highest power in the polynomial. So, if we have a polynomial \(g(x)=x^4+3x^3-1\), the highest power is 4 and so we know that we have 4 roots.
Special Notes
1. Notice that we did not say that the roots are all real values. From the context, it is natural to assume this but that is not the case. For example, the graph of the polynomial \(y=x^2+1\) never crosses the x-axis but it does have two roots, both complex.
2. Just because the roots exist, does not mean we can find them. In fact, it is sometimes not possible to find them exactly from the equation and we may have to go to our calculuator to approximate them.

Finding the roots of a polynomial is pretty straightforward. The technique requires you to understand the concept of the zero product rule since it is used heavily when calculating the roots. Basically the technique requires us to
1. move all terms to one side of the equal sign, leaving zero on the other side; this may also entail setting the polynomial equal to zero;
2. factor the polynomial;
3. use the zero product rule and solve for the x-values.

In the last step, we may be able to solve for the x-values directly or we may need to approximate the values using our calculator or graphing utility. Here is a quick example.

This takes some practice, so let's work some practice problems.