Numerical Approximation of a Zero: Midpoint Method

What is an exact, versus an approximated, value?

In mathematics, the "exact" value will be one with a "closed form" expression, being an expression that can be written without any trailing dot-dot-dot. For instance, is the exact form of the square root of 2, while 1.414213562... is a decimal approximation. Another example comes from geometry, where π is the exact form of the number which is often approximated as 3.14159 or .

You can always find the exact zeroes of a quadratic equation, because you have a formula; namely, the Quadratic Formula. This formula allows you to find the "exact" roots, even when they involve square roots.

What about cubics, quartics, and higher-degree polynomials?

There do exist formulas for solving cubic and quartic equations exactly. But these formulas are so complicated that you'll likely never see or hear of them, let alone have to figure out how to use them. And there are no general formulas for higher-degree polynomials at all.

(Asides: "The Scandalous History of the Cubic Formula" and "Solving the Quartic with a Pencil")

Why do we need numerical approximation?

For many (most?) polynomials, we simply can't find nice neat solutions. There are no general formulas for solving higher-degree polynomials, so we may not be able to find the exact values of any of the roots of that polynomial. So, to find the roots of these messy higher-degree polynomials, we must instead try to find the approximate the values for those roots.

When you are asked (in an algebra class) to find the zeroes (that is, the x-intercepts, the solutions, or the roots) of a polynomial equation, you will almost always be given a polynomial that is actually solvable in some straightforward manner. The polynomials will be carefully structured so that you can find the exact solutions.

But most polynomials (in "real life") aren't solvable. How do you find their zeroes? You don't, at least not exactly. Instead, you find a numerical approximation to some degree of accuracy. (This "to some degree of accuracy" answer is what you get when you punch buttons on your graphing calculator to find the root of a graphed equation.) How does this work?

What is the continuity property of polynomials?

The "continuity property" of polynomials says that, if your polynomial equals, say, 5 at some value of x and equals, say, 10 at some other value of x, then the polynomial takes on every value between 5 and 10 because polynomials graph as continuous (that is, as connected) lines. They don't have gaps or jumps or angles or elbows; they don't do anything hinky or unexpected; they're just nice smooth solid connected curvy lines.

How does continuity help with numerical approximation?

If your polynomial is negative (that is, if its graph is below the x-axis) at one spot and positive (that is, above the x-axis) at another spot, then it has to be true that the polynomial takes on the value of zero somewhere in between those two x-values, because zero falls between the negatives and the positives.

For example, let's look at the polynomial function y = x⁵ + x³ − 3x − 2:

You can quickly see from the graph (below) that the polynomial is negative (below the axis) at x = 1 and positive (above the axis) at x = 2. (Specifically, for x = 1, y = −3; for x = 2, y = 32.) Then the polynomial must be zero (that is to say, it must cross the x-axis) somewhere in between x = 1 and x = 2. To find the zero, you would start looking inside this interval.

Note: In practice, you'll probably be given x-values to use as your starting points, rather than having to find them yourself from a graph. In these cases, one of the x-values will give a negative value for the polynomial and the other will give a positive value. This is similar to when you use your calculator to find zeroes on a graph, and the calculator asks you to pick left- and right-hand bounds for the zero.

There are two methods for narrowing in on a polynomial's zero: using midpoints, and picking clever values for x. I'll show midpoints first.

What is the method of midpoints?

The method of midpoints is a process for finding x-intercepts (that is, of finding real-number solutions to equations) that uses interval midpoints to narrow down the interval in which the zero lies. By making the interval containing the solution half the size at each step, you can get closer and closer to the actual value of the intercept.

To use the method of midpoints, you will usually (in the class where you're studying the topic) be given a beginning interval. In "real life" (such as other classes, labs, etc, where you need to approximate zeroes), you'll have to find that beginning interval yourself. If you have to find the beginning interval yourself, you'll need to find an x-value for which the function's value (that is, the y-value) is negative, and find another x-value where the function is positive. These two x-values will be the endpoints of your initial interval.

Once you have your beginning interval [a, b], follow these steps:

1. Note the signs of the function's values at each of the endpoints.
2. Find the x-value that is the midpoint between the interval's endpoints; that is, add the two x-values, and divide by 2.
3. Find the value of the function at the interval's midpoint; in particular, find the sign of the function at the midpoint.
4. Note the interval endpoint that has the same sign. Between that endpoint and the midpoint, the function's sign is the same, so there is no zero on this half of the interval.
5. Discard this same-sign half of the interval. The midpoint and the other endpoint now bracket your new interval.
6. Repeat steps (1) through (5), until the values of the endpoints match to whatever is the desired number of decimal places. Once they have, say, the same four digits after the decimal place, you'll take that portion of the endpoints' values as your decimal approximation of the x-intercept.

What is an advantage of the method of midpoints?

The midpoint method will always eventually get you to the value you want. Also, because you're always using the interval's midpoint, the process is easy to program. (For instance, if you're in a computer-science class and need to write a program to find x-intercepts, given a starting interval, this method of "add, and then divide by 2, is very easy to program.)

What is a disadvanage of the midpoint method?

If you have a graphing utility, so you can see the graph of the function, you can almost always zoom in on the x-intercept and pick better values than the midpoint. The midpoint method will always work, but it can take you a lot longer, especially if you're working "by hand".

The following is an example of finding a decimal approximation of an x-intercept by using the method of midpoints.

• Given the polynomial x⁵ + x³ − 3x − 2, find its zero on the interval 1 ≤ x ≤ 2, accurate to three decimal places.