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We learned what a Polynomial is here in the Introduction to Multiplying Polynomials section. As a review, here are some polynomials, their names, and their degrees.  Remember that the degree of the polynomial is the highest exponent of one of the terms (add exponents if there are more than one variable in that term). We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively. There are certain rules for sketching polynomial functions, like we had for graphing rational functions. Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form).   The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree).
So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent).
Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent).
Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value). Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root. Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape. Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if    is a root, then so is  .  The complex form of this theorem states that if    is a root, then so is  . Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them.  If we do this, we may be missing solutions! Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots.
For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle! Remember that if we divide a polynomial by “x – c” and get a remainder of 0,  then “x – c” is a factor of the polynomial and “c” is a root. When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start!   In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function  with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are  where p are all the factors of d above, and q are all the factors of a above.
Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The rational root test help us find initial roots to test with synthetic division,  or even by evaluating the polynomial to see if we get 0.  However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above. Now let’s try to find roots of polynomial functions without having a first root to try.
Also remember that you may end up with imaginary numbers as roots, like we did with quadratics.
Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots.  For example, if you take an x out, you’ll add a root of “0”.
If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division.   If you don’t have a calculator, guess a possible rational zero using the  method above. There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root. There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand.  Yes, and it was named after a French guy! The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial  by looking at the sign changes of the terms of that polynomial. We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial. In this section we'll focus on how to sketch the graph of any polynomial function, a really important skill.
There are basically three things to do in order to sketch the graph of a polynomial function. Remember that you have many methods at your disposal: GCF, grouping, idendification of special forms, and the rational root theorem. In the previous section we discussed several ways of finding the roots of polynomial functions. When the independent variable increases in size in either direction ( A± ), the ends of a polynomial graph will eventially increase or decrease without bound (infinitely). End behavior is a clue about the shape of a polynomial graph that you just can't do without, so you should either memorize these possibilities or (better yet) understand where they come from.
The table below also shows that a polynomial function of degree n can have at most n - 1 points where it changes direction from down-going to up-going. The end behavior is down on the left and up on the right, consistent with an odd-degree polynomial with a positive leading coefficient. It takes a few tries to get the hang of this kind of curve sketching, but it will develop with practice. Likewise there are no other options, given the right-end behavior, for the part of f(x) between 0 and 3. What we don't know from such a sketch is just exactly how high the maxima rise and how low the minima dive.
The key to sketching a function like this quickly is seeing that it's just the parent function of all cubic functions, y = x3, shifted to the right by 2 units and inverted across the x-axis. For these kinds of graphs, I like to lightly sketch in the parent function, then apply the transformations one at a time. Sometimes you won't find a GCF, grouping won't work, your function is not a sum or difference of cubes and it doesn't look like a quadratic, .
That means graphing the function on a calculator and estimating x-axis crossings or using a numerical root-finding algorithm. Still, you should be aware of end behavior, the y-intercept and maybe a few other things in order to check that computer output is what you expect it to be.
Polynomial functions are functions of a single independent variable, in which that variable can appear more than once, raised to any integer power.
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.
The appearance of the graph of a polynomial is largely determined by the leading term – it's exponent and its coefficient. Each algebraic feature of a polynomial equation has a consequence for the graph of the function. A solution of f(x) = 0 where the graph just touches the x-axis and turns around (creating a maximum or minimum - see below). 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.
When a graph turns around (up to down or down to up), a maximum or minimum value is created.



The parabola f(x) = x2 has a global minimum at x = 0, but no global maximum (it increases without bound). 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).
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.
In general, we say that the graph of an nth degree polynomial has (at most) n-1 turning points.
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.
Now the solutions to this equation are just the roots or zeros of the polynomial function f(x) = 4x4 - 3x3 + 6x2 - x - 12.
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. Further, when a polynomial function does have a complex root with an imaginary part, it always has a partner, its complex conjugate.
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. In this form, there is a constant term, and the first term has twice the degree as the middle term.
Once you get the hang of this, you won't have to use the substitution trick, but it does help to keep things straight. While this method of finding roots isn't used all that often, it's a huge timesaver when it can be used. You should confirm these formulae for yourself by multiplying and simplifying the right sides. 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. The important thing to keep in mind about the rational root theorem is that any given polynomial may not even have any rational roots. Synthetic substitution is simply a method for substituting a value into the independent variable of a polynomial function. Sometimes (erroneously) called synthetic division, this procedure is illustrated by example on the left. The method starts with writing the coefficients of the polynomial in order of the power of x that they multiply, left to right. The number to be substituted for x is written in the square bracket on the left, and the first coefficient is written below a line drawn under everything (second step). Now we don't want to try another positive root because the coefficients of the new cubic polynomial are all positive.
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, . End Behavior refers to the behavior of a graph as it approaches either negative infinity, or positive infinity.
The coefficient of the term with the highest degree (greatest power of x) is called the leading coeficient and cannot equal 0. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n zeroes, not all of which are necessarily real or unique.
Knowing the multiplicity of a root is useful in determing features of a polynomial's graph, as will be discussed in the next section. The end behavior of any given polynomial is identical to the end behavior of its parent polynomial, .
As can be seen in the graphs above, polynomials have one kind of end behavior when they are of odd degree and another when they are of even degree. For all , the graph grows towards positive infinity as one moves right along the x-axis; that is, . To determine the degree of the polynomial and its leading coefficient, we must multiply together the greatest powers of x in each term of the polynomial.
Problem: For each of the polynomials below, determine the roots and whether the graph passes through or bounces off the axis at each point. Once you have determined the end behavior, y-intercept, x-intercepts, and behavior at x-intercepts for a polynomial, you can easily sketch a graph showing these important features, as shown in the example below. Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”. If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!)  And remember that if you sum up all the multiplicities of the polynomial, you will get the degree! Anybody can plot a function on a computer, but you'll need to know what you're doing to see if the computer has done what you think you asked it to. The fourth item in this list is just a reminder that the graphs of polynomial functions are continuous with no breaks or sharp points of any kind.
We can use a computer to draw a more precise and more accurate graph, but we need a sketch to make sure that graph looks right - that we asked the computer to do the right thing. Think of it as xA·xA·x = 0, and that if either of the three x's are zero, then the whole function has a zero value. This is a double root, which means that the graph of this function just touches the x-axis at x = -4.
In the previous section we showed that the end behavior depends on the sign of the leading coefficient and on the degree of the polynomial.
Notice that all three roots are single roots, so the function graph has to pass right through the x-axis at those points (and no others).
You can see that it has all of the essential features of our sketch, but that the details are filled in.
The ends of this function both go in the same direction because its degree is even, and that direction is upward because the coefficient of the leading term, x4, is positive. You can see that all of the essential features of our sketch were correct; we just have to blow up the region in green to see the other 3 roots (1 double, 1 single).
They work the same way every time, and knowing how they affect a known function will really help you visualize the transformed function. If we can identify the function as just a series of transformations of some parent function that we know, the graph is pretty easy to visualize. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 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.
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, the quadratic function f(x) = (x+2)(x-4) has single roots at x = -2 and x = 4. 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.


The curvature of the graph changes sign at an inflection point between concave-upward and concave-downward.
Pay attention to how the graph behaves at the left and right ends and to how many wiggles it has. We automatically know that x = 0 is a zero of the equation because when we set x = 0, the whole thing zeros out.
First find common factors of subsets of the full polynomial, say two or three terms, and move that out as a common factor.
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). This kind of substitution is also used in all kinds of other situations in algebra, so it's a good idea to learn it.
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. In those cases, we have to resort to estimating roots using a computer, using methods you will learn in calculus. It's a quick and easy method to test whether a value of the independent variable is a root. Now synthetic substitution gives us a quick method to check whether those possibilities are actually roots. Polynomials of degrees 2, 3, and 4 are known as quadratic, cubic, and quartic functions, respectively. Knowing how to find the zeroes of a polynomial is essential for graphing both polynomial and rational functions, and will be necessary for maximization problems in calculus.
Since the product of 0 and anything is 0, the roots are found by setting each term in factored form equal to 0. We are interested in being able to create graphs of polynomials that show their end behavior, y-intercept, x-intercepts, and behavior at roots.
The end behavior of this parent polynomial is determined by the parity of the power to which x is raised (whether it is even or odd) and the sign of the leading coefficient. As one travels left along the x-axis, the graph of goes towards negative infinity if n is odd and towards positive infinity if n is even; symbolically, if n is odd, while if n is even. The cases examined thus far have all had positive leading coefficients (in fact, the leading coefficient has been 1 in each case).
Thus, we find that the first term in the expanded form of the polynomial is , which shows that the polynomial is of degree 4 (even) and has a negative leading coefficient. Thus, x-intercepts on a graph are found exactly as described above in the section Finding Roots. The next logical thing to wonder is what it does at the points immediately around these x-intercepts. But the graph of y=x2 bounces off the axis at the root, while the graph of y=x3 passes through. Thus, the graph will pass through the axis at the root x=1 and bounce off the axis at the root x=-2. Setting the function equal to 0, one finds the roots are x=-2 and x=-3, which have multiplicities of 3 and 2, respectively. A triple root at x = 0 means that there is an inflection point there, a point where the curvature of the function changes between concave-upward and concave-downward. This function doesn't have an inflection point on the x-axis (it may have one or more elsewhere, but we won't be able to find those until we can use calculus). The curve has to smoothly pass right through both points on the x-axis and go to -∞ on the left. Because we've already sketched the graph, we can be confident that the computer output is reliable. Here it is in one sketch with some explanations, but the process goes like this: Draw in the roots, then the end behavior. For example, in f(x) = 8x4 - 4x3 + 3x2 - 2x + 22, as x grows, the term 8x4 dominates all other terms. 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.
The graph of f(x) = x4 is U-shaped (not a parabola!), with only one turning point and one global minimum. Notice also the relative sizes of the effects of changing A-E: Changing A, the coefficient of x5, alters the look of the graph dramatically, while changing the linear parameter E has only a small effect. 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. In the example, if there were no linear term, we'd put 0 in instead of a 1 in the first step. The numbers now aligned in the first and second row are added to become the next number under the line. 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. 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. Thus, the graph will bounce off the axis at the root x=-2 and pass through the axis at the root x=-3. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. The theorem says they're complex, and we know that real numbers are complex numbers with a zero imaginary part.
We haven't simplified our polynomial in degree, but it's nice not to carry around large coefficients. The coefficient of the highest degree term (x4), is one, so its only integer factor is q = 1. 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. 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. If a root has an even multiplicity, the graph will bounce off the axis, as with the graph of y=x2. Well, you're stuck, and you'll have to resort to numerical methods to find the roots of your function. If a root has an odd multiplicity, the graph will pass through the axis, as with the graph of y=x3.



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