Graphing and Factoring Polynomials

The Microworld…

The Story…

Now, let us turn to polynomial functions that are determined by 4 points. Change the points entry to 4:

And follow the procedure described in the previous section. This time, when you have the pointing finger, left-click 4 times at points with distinct x coordinates. You will see something like:

Notice the way the graph "wiggles" It crosses the x-axis (in this case) in 3 points. This is the largest number of times it can cross any horizontal line. It also has two "bumps" that are reminiscent of vertices. Polynomials of degree 3 such as this one may or may not have such bumps in their graphs. But the "concavity" will always change at a single point between being "concave up" to "concave down" or vice-versa. This special point is called the inflection point.

We will see on the Inflection Points and Roots page that the abscissa of the inflection point is the average value of the three roots if the polynomial equation has 3 roots. The graphs of these functions must cross the x axis at least once, but may not cross fully three times. And notice the algebraic form of the function. It is now a sum of 4 terms. It has a constant term, a term that is a constant multiple of , a term that is a constant multiple of  and a term that is a constant multiple of   (in this example, that last constant is  ). The largest power of the variable t is 3 for degree-3 polynomials. That is, they may contain a  term, but may not have a  term.

These degree-3 polynomials also have a colloquial name. They are called cubic polynomials. The shapes of their graphs can be quite varied, but they have the basic property that the graph can cross a horizontal line in at most 3 points. And of course, they are determined by any 4 of their points.

To get an idea of the variation possible in the shapes of cubic graphs, do the following experiment. Construct graphs that pass through the points ,   and   and select for the fourth point, one at a time, each of: ,  ,  ,  , and.  Next, construct the graph through points: , ,  , and . This example has no "bumps".  For this last one, see what the effect is if you keep 3 points the same, but vary one point a little bit. For fun, see if you can find a cubic function that crosses the x-axis exactly 2 (not 3) times. If you have difficulty doing that by clicking points, you may use the command line to define, then graph your function. Just press Enter after each line.

Now, let's consider polynomial functions of degree 4. These are sometimes referred to as quartic polynomials. Type 5 in the "Points" field

Then left-click on the Graph2D window, and then left-click 5 times. You will see something like:

There are 5 terms in the polynomial expression for f(t):

and the coefficient of the  term is . There are now 3 "bumps" in this graph, and the graph crosses the x-axis in 4 places. The graph of a quartic polynomial can cross a horizontal line in at most 4 points. It may cross in fewer points, or it may not cross at all. It is difficult to describe in words the full range of shapes that graphs of quartic functions may have. Use the graphing utility to explore the possibilities of these, and of the graphs of polynomial functions up to degree 9. The basic facts are:

1.      A polynomial function   of degree   is a sum of terms, where each term has the form  with  between 0 and  . If , we usually write

2.      A polynomial of degree   is determined by  of its points.

3.      A polynomial of degree   can cross a horizontal line at most  times.

Polynomial equations:

As you may know, it is often necessary to solve equations of the form: , where   is a polynomial function of . If the degree of f is 1, this reduces to a linear equation:  with obvious solution   in the case that  is not 0. This solution is the unique point on the x-axis that the line crosses if   is not 0. If the degree of f is 2, then this reduces to a quadratic equation

with solutions

given by the quadratic formula on completion of the square.

Now if  < 0, it means the graph of f does not cross the x axis. If  it means the graph of f touches the x-axis at only one point, that is,  it is tangent to it. If  > 0, then the graph of f crosses the x-axis in two points. In any case, the line  is the axis of symmetry.

In general, for a polynomial of degree : , the solutions to the equation  correspond to the points where the graph of f crosses the x-axis. This can be, as we observed in at most  points. But what are those points? That is in general a difficult question to answer by exact means.

On the next page, we will give numerical approximations to the roots of such equations. But while there are exact methods for solving cubic and quartic equations, they are difficult to implement. They would involve expressing the solutions of the polynomial equations in terms of the coefficients of the various powers of t by means of a formula analogous to the quadratic formula. To give an idea how complicated this might be, let us consider the problem of solving a cubic equation.

The solutions of   are given by the cubic formula:

If we let   then the expression

yields the 3 roots, where we take all combinations of square and cube roots (including complex roots).  It is a deep result of the theory of equations that at most 3 distinct numbers will be so found.

For example, if a = b = 0, the equation is  and the formula reduces to:

If a = 0, the equation is  and the formula reduces to:

Following Cardano and Tartaglia, we will learn how to “complete the cube” and will derive this fascinating formula in this book. There is, however, a method that may sometimes be employed to solve polynomial equations of arbitrary degree.

Factoring Polynomial expressions:

If f is a polynomial function of degree : we say that  is a polynomial expression. For example,  is a polynomial expression of degree 3, a cubic expression. As an expression, it may be manipulated according to the rules of algebra in its own right. In particular, it may sometimes be possible to write it as a product (not a sum) of lower-degree polynomials.

Let's look at an example. Suppose we were given the quadratic equation:.  We might use the quadratic formula above to find the two solutions for the equation. The graph in any case would tell us that there are two solutions:

But another way to solve the equation -- to get the exact solutions -- is to factor the quadratic expression:

and to observe that  is a solution to   only if either   or  , that is, only if   or . This algebraic approach to solving quadratic equations extends to all polynomial equations.

Let us see how it works for cubic equations. Construct a cubic polynomial by typing 4 in the Points field:

and by clicking on ,   and any two other points. This will guarantee that x = 2 and x = -3 are solutions of the cubic equation that results from setting the cubic expression defining   to 0. After that, press the “Factor Expression” button. You see something like:

In this case, the polynomial expression reported was:

Of course, it crosses the x axis at x = 2 and x = -3. But it crosses once again at another point. Just below the polynomial expression, there now appears a factorization:

since we pressed Factor Expression. This tells us what the other root is. It is obtained when   or when .

In general, if you select   points on the x-axis for a degree   polynomial, and factor the resulting polynomial expression you will get a complete factorization into n degree 1 polynomials, and the last root of the resulting equation may be found in this way.

You should try a few examples of your own. But it may happen for high-degree polynomials that you will give the system a problem that will take a very long time to factor. If the hourglass cursor sits there for a long time and the system reports no answer, press the Esc key to get control back.

Suppose you had clicked on points:  You would have seen:

This time there are only 2 crossings, only two solutions of the equation :. That's all right. The cubic polynomial still factors into 3 degree-1 terms. The factor  happens to be repeated. Worse things can happen. For example, suppose you had clicked on the points: . You would then have seen:

And in this case, the polynomial reported has only one crossing (and no bumps in the graph). There is no factorization into 3 degree 1 factors. The best we can do is to factor it into a product of a degree 1 factor  and a degree 2 factor . The degree 1 factor yields the root t = -1 but the quadratic factor yields no real roots. This almost covers the range of possibilities for cubic polynomial factorizations. There will always be at least one linear factor, and therefore at least one solution to the resulting cubic equation.

But our Factor Expression Button only finds degree 1 factors with rational number (fraction) coefficients. If the equation has irrational solutions, then the button will not find the factorization, and it will be difficult in general to find it by hand. In this case, we must resort to the cubic formula or to the numerical methods that we'll use on the next page. For polynomial equations of degree 5 or higher, there is no analog of a quadratic or cubic formula, and so there we generally have to resort to approximate numerical methods.

For exercise, find a complete factorization for the polynomials and solve the equation obtained by setting them to 0.

1.

2.

3.

On the Inflection Points and Roots page you will find an experiment that shows the relation between the inflection point and the solutions of a cubic equation, in case it has 3 real solutions.