Definitions & examples

Polynomial functions are functions of a single independent variable, in which that variable can appear more than once, raised to any integer power. For example, the function

is a polynomial. Polynomial functions are sums of terms consisting of a numerical coefficient multiplied by a unique power of the independent variable.

We generally write these terms in decreasing order of the power of the variable, from left to right*. Here is a summary of the structure and nomenclature of a polynomial function:

*Note: There is another approach that writes the terms in order of increasing order of the power of x. This has some appeal because we write power series that way. You'll have to choose which works for you.

Here are some examples of polynomial functions and the language we use to describe them:

f(x) = 3x - 2 Linear polynomial (linear function)
f(x) = x2 - 4x + 1 Quadratic polynomial
f(x) = -3x3 + x - 6 Cubic polynomial with no quadratic term
f(x) = (x - 3)2(2x - 1) Cubic polynomial (convince yourself that the largest power will be three when expanded)
f(x) = 2x4 - 5x3 + 2x2 - x + 17 Quartic polynomial
f(x) = 18x5 - 7 Quintic polynomial with only the 5th degree and constant terms.

Anatomy of a polynomial function

Polynomial functions (we usually just say "polynomials") are used to model a wide variety of real phenomena. In physics and chemistry particularly, special sets of named polynomial functions like Legendre, Laguerre and Hermite polynomials (thank goodness for the French!) are the solutions to some very important problems.

It is important that you become adept at sketching the graphs of polynomial functions and finding their zeros (roots), and that you become familiar with the shapes and other characteristics of their graphs.

Graphs of polynomial functions

The appearance of the graph of a polynomial is largely determined by the leading term – it's exponent and its coefficient. Because the leading term has the largest power, its size outgrows that of all other terms as the value of the independent variable grows. For example, in f(x) = 8x4 - 4x3 + 3x2 - 2x + 22, as x grows, the term 8x4 dominates all other terms.

You can check this out yourself by making a quick spreadsheet. Label one column x and fill it with integer values from 1-10, then calculate the value of each term (4 more columns) as x grows. Sum them and add the constant term (22) to find the value of the polynomial. The leading term will grow most rapidly. Here's what I mean:

Graphs of polynomial functions

Each algebraic feature of a polynomial equation has a consequence for the graph of the function. Here is a table of those algebraic features, such as single and double roots, and how they are reflected in the graph of f(x).

Term Definition
Single root A solution of f(x) = 0 where the graph crosses the x-axis. For example, the quadratic function f(x) = (x+2)(x-4) has single roots at x = -2 and x = 4.
Double root A solution of f(x) = 0 where the graph just touches the x-axis and turns around (creating a maximum or minimum - see below). For example, the cubic function f(x) = (x-2)2(x+5) has a double root at x = 2 and a single root at x = -5.
Triple root A solution of f(x) = 0 where the graph crosses the x-axis and the curvature changes sign. See the graph below. For example, the cubic function f(x) = x3 has a triple root at x = 0.
Inflection point The name of the point that is a triple root of a polynomial function. The curvature of the graph changes sign at an inflection point between concave-upward and concave-downward. Not all inflection points are located at triple roots (or even at roots at all), but all triple roots are inflection points located on the x-axis.
y-intercept The solution to f(0); the point where a graph crosses the y-axis, usually a convenient (and very easy-to-find) point to plot when sketching a graph.
Local maximum/minimum When a graph turns around (up to down or down to up), a maximum or minimum value is created. Local maxima or minima are not the highest or lowest points on a graph.
Global maximum/minimum The parabola f(x) = x2 has a global minimum at x = 0, but no global maximum (it increases without bound). The parabola f(x) = -x2 has a global maximum, but no global minimum. The graph below has a global maximum at x = 1. The highest/lowest point on a graph (one may not exist).
End behavior When x is large, either positive or negative, we are concerned with whether the function increases or decreases without bound (it will do one or the other).

Features of a polynomial graph

End behavior: odd degree

Here are the graphs of two cubic polynomials. they differ only in the sign of the leading coefficient.

The leading term of any polynomial function dominates its behavior. When that term has an odd power of the independent variable (x), negative values of x will yield (for large enough |x|) a negative function value, and positive x a positive value.

That is, for large enough | x |,

f(x > 0) > 0

f(x < 0) < 0

The opposite is true when the coefficient of the leading power of x is negative.

Note also in these figures and the ones below that a cubic polynomial (degree = 3) can have two turning points, points where the slope of the curve turns from positive to negative, or negative to positive. The quartic polynomial (below) has three turning points.

In general, we say that the graph of an nth degree polynomial has (at most) n-1 turning points. It may have fewer, however.

End behavior: even degree

When the degree of a polynomial is even, negative and positive values of the independent variable will yield a positive leading term, unless its coefficient is negative. Negative numbers raised to an even power multiply to a positive result:

(-2)(-2) = 4

(-2)(-2)(-2) = -8

(-2)(-2)(-2)(-2) = 16, and so on.

The result for the graphs of polynomial functions of even degree is that their ends point in the same direction for large | x |:

up when the coefficient of the leading term is positive,

down when the coefficient is negative.

Notice that these quartic functions (left) have up to three turning points. A quartic function need not have all three, however. The graph of f(x) = x4 is U-shaped (not a parabola!), with only one turning point and one global minimum.

The table below summarizes some of these properties of polynomial graphs.

End behavior of polynomial function graphs.

Finding Roots or Zeros

Very often, we are faced with finding the solution to an equation like this:

4x4 - 3x3 + 6x2 = x + 12

Such an equation can always be rearranged by moving all of the terms to the left side, leaving zero on the right side:

4x4 - 3x3 + 6x2 - x - 12 = 0

Now the solutions to this equation are just the roots or zeros of the polynomial function f(x) = 4x4 - 3x3 + 6x2 - x - 12. They are the points at which the graph of f(x) crosses (or touches) the x-axis. Our task now is to explore how to solve polynomial functions with degree greater than two. We already know how to solve quadratic functions of all kinds. First, a little bit of formalism:

The fundamental theorem of algebra

Every non-zero polynomial function of degree n has exactly n complex roots.

The fundamental theorem of algebra tells us that a quadratic function has two roots (numbers that will make the value of the function zero), that a cubic has three, a quartic four, and so forth. The number of roots will equal the degree of the polynomial.

But there's a catch: They don't all have to be real numbers. The theorem says they're complex, and we know that real numbers are complex numbers with a zero imaginary part.

Further, when a polynomial function does have a complex root with an imaginary part, it always has a partner, its complex conjugate.

When a polynomial function has a complex root of the form a + bi, a - bi is also a root. Complex roots with imaginary parts always come in complex-conjugate pairs, a ± bi. When the imaginary part of a complex root is zero (b = 0), the root is a real root.

Methods of finding roots

1. Greatest common factor (GCF)

When faced with finding roots of a polynomial function, the first thing to check is whether there is something that can be factored away from all of its terms. Here are some examples:

x4 - 3x3 + x2 - 7x = 0

x(x3 - 3x2 + x - 7) = 0

Factor out an x, which appears in all terms. We automatically know that x = 0 is a zero of the equation because when we set x = 0, the whole thing zeros out. This leaves us with finding the zeros of a simpler polynomial.

3x3 - 27x + 81 = 0

x3 - 9x + 27 = 0

All terms are divisible by three, so get rid of it. Note that the zero on the right makes this very convenient ... the 3 just "disappears". We haven't simplified our polynomial in degree, but it's nice not to carry around large coefficients.

11x3 - 121x2 + 3x - 2 = 0

Tantalizing when you look at the x's, and the 11 and 121, but there is no GCF here.

Always look for a greatest common factor first when working with any polynomial function. Finding one can make things a lot easier.

Practice problems

Find all roots of these polynomial functions by finding the greatest common factor (GCF).

Factoring by grouping

Sometimes factoring by grouping works. The first thing you'll need to check is whether you've got an even number of terms. If it's odd, move on to another method; grouping won't work.

The example below shows how grouping works. First find common factors of subsets of the full polynomial, say two or three terms, and move that out as a common factor.

If what's been left behind is common to all of the groups you started with, it can also be factored away, leaving a product of binomials that are simpler and easier to solve for roots.

The trickiest part of this for students to understand is the second factoring. Look at the example. Between the second and third steps. The binomial (x + 3) is just treated as any other number or variable. It appears in both added terms of the second step, therefore it can be factored out.

An example of factoring by grouping

Practice problems

Find all roots of these polynomial functions by factoring by grouping.

3. Recognizing the form of a quadratic

You have worked with quadratic equations enough to recognize their basic form:

f(x) = Ax2 + Bx + C.

In this form, there is a constant term, and the first term has twice the degree as the middle term. Now consider equations of the form

Notice that each of those equations has the same pattern. All have three terms, the highest power is twice that of the middle term, and each has a constant term (if it didn't, we'd be able to find a GCF). They have the same general form as a quadratic. Here's an example:

Let's find the roots of the quartic polynomial equation,

To do this, we make a simple substitution: Let u = x2, which means that u2 = x4.

Now this quadratic polynomial is easily factored:

Now we can re-substitute x2 for u like this:

Finally, it's easy to solve for the roots of each binomial, giving us a total of four roots, which is what we expect.

Doing these by substitution can be helpful, especially when you're just learning this technique for this special group of polynomials, but you will eventually just be able to factor them directly, bypassing the substitution.

Substitutions like this, sometimes called u-substitution, are very handy in a number of algebra and calculus problems. Don't shy away from learning them. Sometimes they're the only way to solve a problem!

Practice problems

4. Sum or difference of perfect cubes

While this method of finding roots isn't used all that often, it's a huge time saver when it can be used. You don't have to memorize these formulae (you can always look them up), but use them in situations where your polynomial equation is a sum or difference of cubes, such as

Formulae: Sum or difference of perfect cubes

x3 + y3 = (x + y)(x2 - xy + y2)

x3 - y3 = (x - y)(x2 + xy + y2)

Practice problems

Use the sum/difference of perfect cubes formulae (box above) to find all of the roots (zeros) of these functions:

5. The rational root theorem

The rational root theorem is not a way to find the roots of polynomial equations directly, but if a polynomial function does have any rational roots (roots that can be represented as a ratio of integers), then we can generate a complete list of all of the possibilities. Once we've got that, we need to test each one by plugging it into the function, but there are some shortcuts for doing that, too.

The important thing to keep in mind about the rational root theorem is that any given polynomial may not even have any rational roots. In those cases, we have to resort to estimating roots using a computer, using methods you will learn in calculus.

Consider a polynomial equation of the form

Axn + Bxn-1 + . . . + Z = 0,

where A is the coefficient of the leading term and Z is the constant term. Now let p = the set of all possible integer factors of Z, and their negatives, and let q = the set of all possible integer factors of A, and their negatives. The rational root theorem says that if there are any rational roots of the equation (there may not be), then they will have the form p/q. That is, any rational root of the equation will be one of the p's divided by one of the q's.

Rational root theorem

Given a polynomial function Axn + Bxn-1 + Cxn-2 + ... + Z, where A, B, C, ..., Z are constants, let q be all of the positive and negative integer factors of A (the leading coefficient) and let p = all of the positive and negative integer factors of Z (the constant term). Then if there are any rational roots of the function, they are of the form ±p/q for any combination of p's and q's.

For example, given the polynomial function

f(x) = 6x4 - 4x3 + 9x2 + 3,

The set q = ±{1, 2, 3, 6}, the integer factors of 6, and the set p = ±{1, 3}, the integer factors of 3.

Now we can construct the complete list of all possible rational roots of f(x):

Now it's very important that you understand just what the rational root theorem says. It gives us a list of all possible rational roots, and we need to plug those each, in turn, into the function to test whether they are indeed roots. Not all of them can be, and it's entirely possible that none are.

What remains is to test them. Before we do that, we'll take a brief detour and discuss a very easy way to do that, synthetic substitution.

Synthetic Substitution

Here's a step-by-step example of how synthetic substitution works.

Sometimes (erroneously) called synthetic division, this procedure is illustrated by this example. It's a quick and easy method to test whether a value of the independent variable is a root.

The method starts with writing the coefficients of the polynomial in decreasing order of the power of x that they multiply, left to right. It's important to include a zero if a power of x is missing. In the example, if there had been no linear term, we'd put a 0 in the top line instead of a 1 in the first step.

The number to be substituted for x is written in the square bracket on the left, and the first coefficient is written below the line (second step). That's the setup. Now it's just a matter of doing the same thing to the end.

The number in the bracket is multiplied by the first number below the line. The result becomes the next number in the second row, above the line. The numbers now aligned in the first and second row are added to become the next number under the line. Repeat until you're finished. The last number below the line is the result of substituting the value in the bracket into f(x).

In our example, -1 is a root because it makes the function zero. The binomial (x+1) must then be a factor of f(x).

The rational root theorem gives us possibilities of rational roots, if any exist. Now synthetic substitution gives us a quick method to check whether those possibilities are actually roots. Using the rational root theorem is a trial-and-error procedure, and it's important to remember that any given polynomial function may not actually have any rational roots. Its roots might be irrational (repeating decimals) or imaginary.


Find the four solutions to the equation x4 + 4x3 + 2x2 - 4x - 3 = 0

We begin by identifying the p's and q's. For this function it's pretty easy. The constant term is 3, so its integer factors are p = 1, 3. The coefficient of the highest degree term (x4), is one, so its only integer factor is q = 1. Therefore our candidates for rational roots are:

rat root 1.1

Now we test to see if any of these is a root. For work in math class, here's a hint: always try the smallest integer candidates first. This is just a matter of practicality; some of these problems can take a while and I wouldn't want you to spend an inordinate amount of time on any one, so I'll usually make at least the first root a pretty easy one. Here we try one and see that it's a root because the value of the function is zero. Notice that the coefficients of the new polynomial, with the degree dropped from 4 to 3, are right there in the bottom row of the synthetic substitution grid.

rat root 1.2

Now we don't want to try another positive root because the coefficients of the new cubic polynomial are all positive. There's no way that a positive value for x will ever make the function equal zero. We'll try the next-easiest candidate, x = -1:

rat root 1.3

That worked, and now we're left with a quadratic function multiplied by our two factors. That's good news because we know how to deal with quadratics. This one is easily factorable:

rat root 1.4

So that's the whole problem. The complete factorization is:

rat root 1.5

and the roots are

rat root 1.6

Practice problems

Find all of the roots (zeros) of these polynomial functions:

What if none of these techniques work?

Sometimes you won't find a GCF, grouping won't work, it's not a sum or difference of cubes and it doesn't look like a quadratic, . . . and it doesn't have any rational roots. What to do? Well, you're stuck, and you'll have to resort to numerical methods to find the roots of your function.

That means graphing the function on a calculator and estimating x-axis crossings or using a numerical root-finding algorithm. Some calculators and many computer programs can do this. You'll also learn about Newton's method of finding roots in calculus.

Video examples

Example 1:   f(x) = x4 + x3 - 3x2 - x + 2

This function has an odd number of terms, so it's not group-able, and there's no greatest common factor (GCF), so it's a good candidate for using the rational root theorem with the set of possible rational roots: {±1, ±2}. If none of those work, f(x) has no rational roots (this one does, though).

Minutes of your life: 3:02

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