You are often asked to find all the zeros (roots or x-intercepts) of polynomials.
To do this in the most efficient way, use the rational zero test and your graphing calculator.
First, there are some general concepts.
So the leading coefficient a gets into the zero finding game as a divisor. This makes sense in general because to get from to the place where we can use the idea that the rational roots are factors in the constant, we would need to factor out a.
Then . And, .
In general, for a polynomial with integer coefficients,
The ri include the rational roots, irrational roots as irrational conjugate pairs , and the complex zeros as the complex conjugate pairs .
The only ones we are likely to find graphically are the rational zeros. If we don't know how to set up the calculator to help us, we might not even find those!
is the largest possible rational zero.
(Why is this true?)
So, if we use the window where x-min is and x-max is , all rational zeros must be in the window.
While we should "see" all the zeros on the graph, we might confuse rational zeros like x = 2 with an irrational zero like . They are too close together to differentiate easily by graph.
We need a table! If we set the table delta or table pitch to be , we should be able to track them all down.
Finding Rational Zeros
Find all zeros of .
The Rational Zero Test tells us that the possible rational zeros are .
We could test those eight possibilities using synthetic division. However,
I am way too lazy for that!
Synthetic division is the best mechanism for finding the irrational and complex zeros. The next example get's that point across:
Finding Irrational Zeros
Find all zeros of . Notice "all" means rational, irrational and complex.
Look at the graph. You should always refer back to it!
Now we bring out the heavy guns. With the two known zeros we use synthetic division to find the rest.
Finding Complex Zeros
Look at this example. The polynomial and synthetic division are in the box.
Graph the polynomial to confirm the rational zeros. Then click on the polynomial to see the graph. Notice that the graph failed to reach the x-axis. The complex zeros have created a "wiggle" in the graph.
The set up is just like before. However, now we have discovered an irreducible quadratic with complex zeros.
The irreducible quadratic solves to give us zeros at x = .
Using the rational root theorem and the graphing calculator together is a powerful technique. It does fail occasionally. The possibility of both irrational and complex zeros for the polynomial cannot be ignored. These are tough nuts to crack.
When that happens, break out a bigger hammer! We revert to programs such as MathCad®, or Maple®, or Mathematica®. These programs can handle these types of polynomials, but require a little more experience and a computer. They also are not allowed during testing!
© 2011 Terry Turner and the ASU School of Mathematical and Statistical Sciences- All rights reserved.