We explore polynomial functions.
Consider . Is this a polynomial function? We can re-write the formula for as Comparing this with our definition of Polyomial Functions, we identify , , , , , and . In other words, is the coefficient of , is the coefficient of , and so forth; the subscript on the ’s merely indicates to which power of the coefficient belongs.
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- (a)
- We note directly that the domain of is . By definition, a polynomial has all real numbers as its domain. Hence, can’t be a polynomial.
- (b)
- Even though simplifies to , which certainly looks like the form given in our definition of polynomials, the domain of , which, as you may recall, we determine before we simplify, excludes . Alas, is not a polynomial function for the same reason isn’t.
- (c)
- After what happened with in the previous part, you may be a little shy about simplifying to , which certainly fits our definition of polynomial functions. If we look at the domain of before we simplified, we see that it is, indeed, all real numbers. A function which can be written in the form of of a polynomial whose domain is all real numbers is, in fact, a polynomial.
- (d)
- We can rewrite as . Since is not a natural number, is not a polynomial.
- (e)
- The function isn’t a polynomial, since it can’t be written as a combination of powers of even though it can be written as a piecewise function involving polynomials. As we shall see in this section, graphs of polynomials possess a quality (which relies on Calculus to verify) that the graph of does not. Polynomials will all be smooth with no sharp corners.
- (f)
- There’s nothing in our definition of a polynomial which prevents all the coefficients , etc., from being . Hence, , is an honest-to-goodness polynomial.
Suppose is a polynomial function.
- Given with , we say
- The natural number is called the degree of the polynomial .
- If , then we call a term of the polynomial.
- We call the coefficient of the term .
- The term is called the leading term or highest degree term of the polynomial .
- The real number is called the leading coefficient of the polynomial .
- The real number is called the constant term of the polynomial .
- The -intercepts of polynomials are also called roots. Note that we usually reserve the word roots for talking about the -intercepts of polynomials and don’t use it for the -intrecepts of other types of functions.
- If , and , we say has degree .
- If , we say has no degree.
The reader may well wonder why we have chosen to separate off constant functions from the other polynomials. Why not just lump them all together and, instead of forcing to be a natural number, , allow to be a whole number, . We could unify all of the cases, since, after all, isn’t ? The answer is ‘yes, as long as .’ The function and are different, because their domains are different. The number is defined, whereas is not. (Technically, is an indeterminant form, which is a special case of being undefined. You will explore this more in calculus.) Indeed, much of the theory we will develop in this chapter doesn’t include the constant functions, so we might as well treat them as outsiders from the start. One good thing that comes from our definition of polynomials is that we can now think of linear functions as degree (or ‘first degree’) polynomial functions and quadratic functions as degree (or ‘second degree’) polynomial functions.
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- (a)
- There are no surprises with . It matches the form of a polynomial given above, and we see that the degree is , the leading term is , the leading coefficient is and the constant term is .
- (b)
- The form given in has the highest power of first. To that end, we re-write , and see that the degree of is , the leading term is , the leading coefficient is and the constant term is .
- (c)
- We need to rewrite the formula for so that it resembles the form given in our definition of polynomials: . The degree of is , the leading term is , the leading coefficient is and the constant term is .
- (d)
- It may seem that we have some work ahead of us to get in standard form. However, it is possible to glean the information requested about without multiplying out the entire expression . The leading term of will be the term which has the highest power of . The way to get this term is to multiply the terms with the highest power of from each factor together - in other words, the leading term of is the product of the leading terms of the factors of . Hence, the leading term of is . This means that the degree of is and the leading coefficient is . As for the constant term, we can perform a similar trick. The constant term is obtained by multiplying the constant terms from each of the factors .
End Behavior of Polynomials
The end behavior of a function is a way to describe what is happening to the function values (the -values) as the -values go off the graph on the left and right sides. That is, what happens to as becomes large (in the sense of its absolute value) and negative without bound (written ) and, on the flip side, as becomes large and positive without bound (written ).
For example, given , as , we imagine substituting , , etc., into to get , , and so on. Thus the function values are becoming larger and larger positive numbers (without bound). To describe this behavior, we write: as , . If we study the behavior of as , we see that in this case, too, .
The same can be said for any function of the form where is an even natural number. For example, the functions , , and are graphed below.
Suppose where is a real number and is an even natural number. The end behavior of the graph of matches one of the following:
- for , as , and as ,
- for , as , and as ,
Graphically:
As with the even degreed functions we studied earlier, we can generalize their end
behavior.
Suppose where is a real number and is an odd natural number. The end behavior of the graph of matches one of the following:
- for , as , and as ,
- for , as , and as ,
Graphically:
To see why this theorem is true, let’s first look at a specific example. Consider . If we wish to examine end behavior, we look to see the behavior of as . Since we’re concerned with ’s far down the -axis, we are far away from so can rewrite for these values of as
As becomes unbounded (in either direction), the terms and become closer and closer to , as the table below indicates.
In other words, as , , which is the leading term of . The formal proof of this theorem requires calculus, but it works in much the same way. Factoring out the leading term leaves
As , any term with an in the denominator becomes closer and closer to , and we have .
Geometrically, this theorem says that if we graph using a graphing calculator, and continue to ‘zoom out’, the graph of it and its leading term become indistinguishable. Below are the graphs of (the thicker line) and (the thinner line) in two different windows.
Other Properties of Polynomial Graphs
Despite having different end behavior, all functions of the form for natural numbers share two special function properties: they are continuous and smooth. While these concepts are formally defined using Calculus, informally, graphs of continuous functions have no ‘breaks’ or ‘holes’ in them, and the graphs of smooth functions have no ‘sharp turns’. It turns out that these traits are preserved when functions are added together, so general polynomial functions inherit these qualities. Below we find the graph of a function which is neither smooth nor continuous, and to its right we have a graph of a polynomial, for comparison. The function whose graph appears on the left fails to be continuous where it has a ‘break’ or ‘hole’ in the graph; everywhere else, the function is continuous. The function is continuous at the ‘corner’ and the ‘cusp’, but we consider these ‘sharp turns’, so these are places where the function fails to be smooth. Apart from these four places, the function is smooth and continuous. Polynomial functions are smooth and continuous everywhere, as exhibited in the graph on the right.
The notion of smoothness is what tells us graphically that, for example, , whose
graph is the characteristic ‘’ shape, cannot be a polynomial.
Roots of Polynomials
We will often what to find the -intercepts or roots of polynomials. To do this, we will use the fact that a product of factors can only equal 0 if one of the factors equals zero. Consider the following example.
You may notice that this polynomial was given as a product of linear factors and irreducible quadratic factors. This was not a coincidence. We have the following theorem.
This way of writing polynomials is extremely helpful when you want to find the roots. Its is so helpful we give it a name.
It turns out that that is a root of the polynomial exactly when is a factor that appears when is written in Factored Form, just like in our example above.
You may noticed another phenomena in our earlier example. Some factors were raised to a power higher than 1. For example, in , the factor is raised to a the second power. This does not mean we get any additional roots. Notice that and so both of these factors give a root of . Instead, the second power has an impact on the way the graph of the polynomial, , looks when it is near the root . On either side of , the graph of will bounce off the -axis and turn around just like the graph of does as the point . In general, it will turn out that if we have in the factored form of our polynomial, then the graph of the polynomial near will look similar to how the graph of looks near .
We give this concept a name and solidify it with a theorem.
Hence, rewriting as , we see that is a zero of multiplicity , is a zero of multiplicity and is a zero of multiplicity .
- If is even, the graph of touches and rebounds from the -axis at .
- If is odd, the graph of crosses through the -axis at .
Graphing Polynomials
Our last example shows how end behavior and multiplicity allow us to sketch a decent graph without appealing to a sign diagram.
Note that we don’t yet know what is happening in that dotted region in the middle.
Next, we find the zeros of . Fortunately for us, is factored. Setting each factor equal to zero gives is and as zeros.
To find the multiplicity of we note that it corresponds to the factor . This isn’t strictly in Factored Form. If we factor out the , however, we get , and we see that the multiplicity of is . Since is an odd number, we know from our theorem about multiplicity that the graph of will cross through the -axis at . What’s more, we know that the graph will pass right through the -axis at without flattening out because the graph will look similar to the way looks at when we zoom in around .
Since the zero corresponds to the factor , we find its multiplicity to be which is an even number. As such, the graph of will touch and rebound from the -axis at .
Though we’re not asked to, we can find the -intercept by finding . Thus is an additional point on the graph. Putting this together gives us the graph below.
Note that while we can say a lot about how graphs of polynomials will look, we cannot say exactly where the turning points (peaks and valleys) will be. That will require calculus.
Factoring
The following section will contain some techniques that are useful for factoring polynomials.
A common technique is factoring out a common factor. For example, in the polynomial , each term contains a factor of . Therefore, we can factor it out to find .
To factor a polynomial like we first multiply together the leading coefficient and the constant term to find 10. Our goal is then to find factors of 10 that sum to the linear coefficient, which in this case is -11. and fit the bill, since and . We can now replace the linear term with . This yields We can then group the terms by two: Factoring out a common factor from each yields Now, we factor out a common factor of from each term, yielding This completes the factorization.
We can also apply this idea of factoring by grouping to higher-degree polynomials. Given the polynomial we can group the terms by two: Factoring out a common factor from each yields Now, we factor out a common factor of from each term, yielding Note that this further factors as .
To factor a difference of squares , we can use the formula which can be checked by multiplying out the right-hand side.
To factor a difference of cubes, , we can use the formula which can similarly be checked by multiplying out the right-hand side.
To factor a sum of cubes, , we can use the formula