5.2: Polynomials
- What properties of a polynomial function can we deduce from its algebraic structure?
- What is a sign chart and how does it help us understand a polynomial function's behavior?
- How do zeros of multiplicity other than \(1\) impact the graph of a polynomial function?
We know that linear functions are the simplest of all functions we can consider: their graphs have the simplest shape, their average rate of change is always constant (regardless of the interval chosen), and their formula is elementary. Moreover, computing the value of a linear function only requires multiplication and addition.
If we think of a linear function as having formula \(L(x) = b + mx\text{,}\) and the next-simplest functions, quadratic functions, as having form \(Q(x) = c + bx + ax^2\text{,}\) we can see immediate parallels between their respective forms and realize that it's natural to consider slightly more complicated functions by adding additional power functions.
Indeed, if we instead view linear functions as having form
(for some constants \(a_0\) and \(a_1\)) and quadratic functions as having form
(for some constants \(a_0\text{,}\) \(a_1\text{,}\) and \(a_2\)), then it's natural to think about more general functions of this same form, but with additional power functions included.
Given real numbers \(a_0, a_1, \ldots, a_n\) where \(a_n \ne 0\text{,}\) we say that the function
is a polynomial of degree \(n\) . In addition, we say that the values of \(a_i\) are the coefficients of the polynomial and the individual power functions \(a_i x^i\) are the terms of the polynomial. Any value of \(x\) for which \(P(x) = 0\) is called a zero of the polynomial.
The polyomial function \(P(x) = 3 - 7x + 4x^2 - 2x^3 + 9x^5\) has degree \(5\text{,}\) its constant term is \(3\text{,}\) and its linear term is \(-7x\text{.}\)
Since a polynomial is simply a sum of constant multiples of various power functions with positive integer powers, we often refer to those individual terms by referring to their individual degrees: the linear term, the quadratic term, and so on. In addition, since the domain of any power function of the form \(p(x) = x^n\) where \(n\) is a positive whole number is the set of all real numbers, it's also true the the domain of any polynomial function is the set of all real numbers.
Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zy . There you'll find a degree \(4\) polynomial of the form \(p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\text{,}\) where \(a_0, \ldots, a_4\) are set up as sliders. In the questions that follow, you'll experiment with different values of \(a_0, \ldots, a_4\) to investigate different possible behaviors in a degree \(4\) polynomial.
-
What is the largest number of distinct points at which \(p(x)\) can cross the \(x\)-axis?
For a polynomial \(p\text{,}\) we call any value \(r\) such that \(p(r) = 0\) a zero of the polynomial. Report the values of \(a_0, \ldots, a_4\) that lead to that largest number of zeros for \(p(x)\text{.}\)
- What other numbers of zeros are possible for \(p(x)\text{?}\) Said differently, can you get each possible number of fewer zeros than the largest number that you found in (a)? Why or why not?
-
We say that a function has a
turning point
if the function changes from decreasing to increasing or increasing to decreasing at the point. For example, any quadratic function has a turning point at its vertex.
What is the largest number of turning points that \(p(x)\) (the function in the Desmos worksheet) can have? Experiment with the sliders, and report values of \(a_0, \ldots, a_4\) that lead to that largest number of turning points for \(p(x)\text{.}\)
- What other numbers of turning points are possible for \(p(x)\text{?}\) Can it have no turning points? Just one? Exactly two? Experiment and explain.
- What long-range behavior is possible for \(p(x)\text{?}\) Said differently, what are the possible results for \(\displaystyle \lim_{x \to -\infty} p(x)\) and \(\displaystyle \lim_{x \to \infty} p(x)\text{?}\)
- What happens when we plot \(y = a_4 x^4\) in and compare \(p(x)\) and \(a_4 x^4\text{?}\) How do they look when we zoom out? (Experiment with different values of each of the sliders, too.)
Key results about polynomial functions
Our observations in Preview Activity 5.2.1 generalize to polynomials of any degree. In particular, it is possible to prove the following general conclusions regarding the number of zeros, the long-range behavior, and the number of turning points any polynomial of degree \(n\text{.}\)
For any degree \(n\) polynomial \(p(x) = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + a_n x^n\text{,}\) has at most \(n\) real zeros. 1
For any degree \(n\) polynomial \(p(x) = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + a_n x^n\text{,}\) its long-range behavior is the same as its highest-order term \(q(x) = a_n x^n\text{.}\) Thus, any polynomial of even degree appears “U-shaped” (\(\cup\) or \(\cap\text{,}\) like \(x^2\) or \(-x^2\)) when we zoom way out, and any polynomial of odd degree appears “chair-shaped” (like \(x^3\) or \(-x^3\)) when we zoom way out.
In Figure \(\PageIndex{4}\) , we see how the degree \(7\) polynomial pictured there (and in Figure \(\PageIndex{3}\) as well) appears to look like \(q(x) = -x^7\) as we zoom out.
Finally, a key idea from calculus justifies the fact that the maximum number of turning points of a degree \(n\) polynomial is \(n-1\text{,}\) as we conjectured in the degree \(4\) case in Preview Activity 5.2.1. Moreover, the only possible numbers of turning points must have the same parity as \(n-1\text{;}\) that is, if \(n-1\) is even, then the number of turning points must be even, and if instead \(n-1\) is odd, the number of turning points must also be odd. For instance, for the degree \(7\) polynomial in Figure \(\PageIndex{3}\) , we know that it is chair-shaped, with one end up and one end down. There could be zero turning points and the function could always decrease. But if there is at least one, then there must be a second, since if there were only one the function would decrease and then increase without turning back, which would force the graph to appear U-shaped.
For any degree \(n\) polynomial \(p(x) = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + a_n x^n\text{,}\) if \(n\) is even, its number of turning points is exactly one of \(n-1\text{,}\) \(n-3\text{,}\) \(\ldots\text{,}\) \(1\text{,}\) and if \(n\) is odd, its number of turning points is exactly one of \(n-1\text{,}\) \(n-3\text{,}\) \(\ldots\text{,}\) \(0\text{.}\)
By experimenting with coefficients in
Desmos
, find a formula for a polynomial function that has the stated properties, or explain why no such polynomial exists. (If you enter
p(x)=a+bx+cx^2+dx^3+fx^4+gx^5
in
Desmos
2
, you'll get prompted to add sliders that make it easy to explore a degree \(5\) polynomial.)
- A polynomial \(p\) of degree \(5\) with exactly \(3\) real zeros, \(4\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
- A polynomial \(p\) of degree \(4\) with exactly \(4\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
- A polynomial \(p\) of degree \(6\) with exactly \(2\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = -\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
- A polynomial \(p\) of degree \(5\) with exactly \(5\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
Using zeros and signs to understand polynomial behavior
Just like a quadratic function can be written in different forms (standard: \(q(x) = ax^2 + bx + c\text{,}\) vertex: \(q(x) = a(x-h)^2 + k\text{,}\) and factored: \(q(x) = a(x-r_1)(x-r_2)\)), it's possible to write a polynomial function in different forms and to gain information about its behavior from those different forms. In particular, if we know all of the zeros of a polynomial function, we can write its formula in factored form, which gives us a deeper understanding of its graph.
The Zero Product Property states that if two or more numbers are multiplied together and the result is \(0\text{,}\) then at least one of the numbers must be \(0\text{.}\) We use the Zero Product Property regularly with polynomial functions. If we can determine all \(n\) zeros of a degree \(n\) polynomial, and we call those zeros \(r_1\text{,}\) \(r_2\text{,}\) \(\ldots\text{,}\) \(r_n\text{,}\) we can write
Moreover, if we are given a polynomial in this factored form, we can quickly determine its zeros. For instance, if \(p(x) = 2(x+7)(x+1)(x-2)(x-5)\text{,}\) we know that the only way \(p(x) = 0\) is if at least one of the factors \((x+7)\text{,}\) \((x+1)\text{,}\) \((x-2)\text{,}\) or \((x-5)\) equals \(0\text{,}\) which implies that \(x = -7\text{,}\) \(x = -1\text{,}\) \(x = 2\text{,}\) or \(x = 5\text{.}\) Hence, from the factored form of a polynomial, it is straightforward to identify the polynomial's zeros, the \(x\)-values at which its graph crosses the \(x\)-axis. We can also use the factored form of a polynomial to develop what we call a sign chart , which we demonstrate in Example 5.2.5.
Consider the polynomial function \(p(x) = k(x-1)(x-a)(x-b)\text{.}\) Suppose we know that \(1 \lt a \lt b\) and that \(k \lt 0\text{.}\) Fully describe the graph of \(p\) without the aid of a graphing utility.
- Answer
-
Since \(p(x) = k(x-1)(x-a)(x-b)\text{,}\) we immediately know that \(p\) is a degree \(3\) polynomial with \(3\) real zeros: \(x = 1, a, b\text{.}\) We are given that \(1 \lt a \lt b\) and in addition that \(k \lt 0\text{.}\) If we expand the factored form of \(p(x)\text{,}\) it has form \(p(x) = kx^3 + \cdots\text{,}\) and since we know that when we zoom out, \(p(x)\) behaves like \(kx^3\text{,}\) we know that with \(k \lt 0\) it follows \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
Since \(p\) is degree \(3\) and we know it has zeros at \(x = 1, a, b,\) we know there are no other locations where \(p(x) = 0\text{.}\) Thus, on any interval between two zeros (or to the left of the least or the right of the greatest), the polynomial cannot change sign. We now investigate, interval by interval, the sign of the function.
When \(x \lt 1\text{,}\) it follows that \(x - 1 \lt 0\text{.}\) In addition, since \(1 \lt a \lt b\text{,}\) when \(x \lt 1\text{,}\) \(x\) lies to the left of \(1\text{,}\) \(a\text{,}\) and \(b\text{,}\) which also makes \(x-a\) and \(x-b\) negative. Moreover, we know that the constant \(k \lt 0\text{.}\) Hence, on the interval \(x \lt 1\text{,}\) all four terms in \(p(x) = k(x-1)(x-a)(x-b)\) are negative, which we indicate by writing “\(----\)” in that location on the sign chart pictured in Figure \(\PageIndex{6}\)
In addition, since there are an even number of negative terms in the product, the overall product's sign is positive, which we indicate by the single “\(+\)” beneath “\(----\)”, and by writing “POS” below the coordinate axis.
We now proceed to the other intervals created by the zeros. On \(1 \lt x \lt a\text{,}\) the term \((x-1)\) has become positive, since \(x \gt 1\text{.}\) But both \(x-a\) and \(x-b\) are negative, as is the constant \(k\text{,}\) and thus we write “\(-+--\)” for this interval, which has overall sign “\(-\)”, as noted in the figure. Similar reasoning completes the diagram.
From all of the information we have deduced about \(p\text{,}\) we conclude that regardless of the locations of \(a\) and \(b\text{,}\) the graph of \(p\) must look like the curve shown in Figure \(\PageIndex{7}\)
Consider the polynomial function given by
- What is the degree of \(p\text{?}\) How can you tell without fully expanding the factored form of the function?
- What can you say about the sign of the factor \((x^2 + 10000)\text{?}\)
- What are the zeros of the polynomial \(p\text{?}\)
- Construct a sign chart for \(p\) by using the zeros you identified in (c) and then analyzing the sign of each factor of \(p\text{.}\)
- Without using a graphing utility, construct an approximate graph of \(p\) that has the zeros of \(p\) carefully labeled on the \(x\)-axis.
- Use a graphing utility to check your earlier work. What is challenging or misleading when using technology to graph \(p\text{?}\)
Multiplicity of polynomial zeros
In Activity 5.2.3 , we found that one of the zeros of the polynomial \(p(x) = 4692(x + 1520)(x^2 + 10000)(x - 3471)^2 (x - 9738)\) leads to different behavior of the function near that zero than we've seen in other situations. We now consider the more general situation where a polynomial has a repeated factor of the form \((x-r)^n\text{.}\) When \((x-r)^n\) is a factor of a polynomial \(p\text{,}\) we say that \(p\) has a zero of multiplicity \(n\) at \(x = r\text{.}\)
To see the impact of repeated factors, we examine a collection of degree \(4\) polynomials that each have \(4\) real zeros. We start with the simplest of all, the function \(f(x) = x^4\text{,}\) whose zeros are \(x = 0, 0, 0, 0\text{.}\) Because the factor “\(x-0\)” is repeated \(4\) times, the zero \(x = 0\) has multiplicity \(4\text{.}\)
Next we consider the degree \(4\) polynomial \(g(x) = x^3 (x-1)\text{,}\) which has a zero of multiplicity \(3\) at \(x = 0\) and a zero of multiplicity \(1\) at \(x = 1\text{.}\)
Observe that in Figure \(\PageIndex{9}\) , the up-close plot near the zero \(x = 0\) of multiplicity \(3\text{,}\) the polynomial function \(g\) looks similar to the basic cubic polynomial \(-x^3\text{.}\) In addition, in Figure \(\PageIndex{10}\) , we observe that if we zoom in even futher on the zero of multiplicity \(1\text{,}\) the function \(g\) looks roughly linear, like a degree \(1\) polynomial. This type of behavior near repeated zeros turns out to hold in other cases as well.
If we next let \(h(x) = x^2 (x-1)^2\text{,}\) we see that \(h\) has two distinct real zeros, each of multiplicity \(2\text{.}\) The graph of \(h\) in Figure \(\PageIndex{11}\) shows that \(h\) behaves similar to a basic quadratic function near each of those zeros and thus shows U-shaped behavior nearby. If instead we let \(k(x) = x^2(x-1)(x+1)\text{,}\) we see approximately linear behavior near \(x = -1\) and \(x = 1\) (the zeros of multiplicity \(1\)), and quadratic (U-shaped) behavior near \(x = 0\) (the zero of multiplicity \(2\)), as seen in Figure \(\PageIndex{12}\)
Finally, if we consider \(m(x) = (x+1)x(x-1)(x-2)\text{,}\) which has \(4\) distinct real zeros each of multiplicity \(1\text{,}\) we observe in Figure \(\PageIndex{13}\) that zooming in on each zero individually, the function demonstrates approximately linear behavior as it passes through the \(x\)-axis.
Our observations with polynomials of degree \(4\) in the various figures above generalize to polynomials of any degree.
If \((x-r)^n\) is a factor of a polynomial \(p\text{,}\) then \(x = r\) is a zero of \(p\) of multiplicity \(n\text{,}\) and near \(x = r\) the graph of \(p\) looks like either \(-x^n\) or \(x^n\) does near \(x = 0\text{.}\) That is, the shape of the graph near the zero is determined by the multiplicity of the zero.
For each of the following prompts, try to determine a formula for a polynomial that satisfies the given criteria. If no such polynomial exists, explain why.
- A polynomial \(f\) of degree \(10\) whose zeros are \(x = -12\) (multiplicity \(3\)), \(x = -9\) (multiplicity \(2\)), \(x = 4\) (multiplicity \(4\)), and \(x = 10\) (multiplicity \(1\)), and \(f\) satisfies \(f(0) = 21\text{.}\) What can you say about the values of \(\lim_{x \to -\infty} f(x)\) and \(\lim_{x \to \infty} f(x)\text{?}\)
- A polynomial \(p\) of degree \(9\) that satisfies \(p(0) = -2\) and has the graph shown in Figure 5.2.14. Assume that all of the zeros of \(p\) are shown in the figure.
- A polynomial \(q\) of degree \(8\) with \(3\) distinct real zeros (possibly of different multiplicities) such that \(q\) has the sign chart in Figure 5.2.15 and satisfies \(q(0) = -10\text{.}\)
5. A polynomial \(q\) of degree \(9\) with \(3\) distinct real zeros (possibly of different multiplicities) such that \(q\) satisfies the sign chart in Figure 5.2.15 and satisfies \(q(0) = -10\text{.}\)
6. A polynomial \(p\) of degree \(11\) that satisfies \(p(0) = -2\) and \(p\) has the graph shown in Figure 5.2.14. Assume that all of the zeros of \(p\) are shown in the figure.
Summary
-
From a polynomial function's algebraic structure, we can deduce several key traits of the function.
-
If the function is in standard form, say
\begin{equation*} p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1}x^{n-1} + a_n x^n\text{,} \end{equation*}
we know that its degree is \(n\) and that when we zoom out, \(p\) looks like \(a_n x^n\) and thus has the same long-range behavior as \(a_n x^n\text{.}\) Thus, \(p\) is chair-shaped if \(n\) is odd and U-shaped if \(n\) is even. Whether \(\lim_{n \to \infty} p(x)\) is \(+\infty\) or \(-\infty\) depends on the sign of \(a_n\text{.}\)
-
If the function is in factored form, say
\begin{equation*} p(x) = a_n(x-r_1)(x-r_2) \cdots (x-r_n) \end{equation*}
(where the \(r_i\)'s are possibly not distinct and possibly complex), we can quickly determine both the degree of the polynomial (\(n\)) and the locations of its zeros, as well as their multiplicities.
-
If the function is in standard form, say
- A sign chart is a visual way to identify all of the locations where a function is zero along with the sign of the function on the various intervals the zeros create. A sign chart gives us an overall sense of the graph of the function, but without concerning ourselves with any specific values of the function besides the zeros. For a sample sign chart, see Figure 5.2.6.
-
When a polynomial \(p\) has a repeated factor such as
\begin{equation*} p(x) = (x-5)(x-5)(x-5) = (x-5)^3\text{,} \end{equation*}
we say that \(x = 5\) is a zero of multiplicity \(3\text{.}\) At the point \(x = 5\) where \(p\) will cross the \(x\)-axis, up close it will look like a cubic polynomial and thus be chair-shaped. In general, if \((x-r)^n\) is a factor of a polynomial \(p\) so that \(x = r\) is a zero of multiplicity \(n\text{,}\) the polynomial will behave near \(x = r\) like the polynomial \(x^n\) behaves near \(x = 0\text{.}\)
Exercises
Consider the polynomial function given by
- What is the degree of \(p\text{?}\)
- What are the real zeros of \(p\text{?}\) State them with multiplicity.
- Construct a carefully labeled sign chart for \(p(x)\text{.}\)
- Plot the function \(p\) in Desmos . Are the zeros obvious from the graph? How do you have to adjust the window in order to tell? Even in an adjusted window, can you tell them exactly from the graph?
-
Now consider the related but different polynomial
\begin{equation*} q(x) = -0.0005(x+21.7)^3 (x-20.9)^2 (x-31.4)(x^2+100)(x-92.3)\text{.} \end{equation*}
What is the degree of \(q\text{?}\) What are the zeros of \(q\text{?}\) What is obvious from its graph and what is not?
Consider the (non-polynomial) function \(r(x) = e^{-x^2}(x^2+1)(x-2)(x-3)\text{.}\)
- What are the zeros of \(r(x)\text{?}\) (Hint: is \(e^{\Box}\) ever equal to zero?)
- Construct a sign chart for \(r(x)\text{.}\)
- Plot \(r(x)\) in Desmos . Is the sign and overall behavior of \(r\) obvious from the plot? Why or why not?
- From the graph, what appears to be the value of \(\lim_{x \to \infty} r(x)\text{?}\) Why is this surprising in light of the behavior of \(f(x)=(x^2+1)(x-2)(x-3)\) as \(x \to \infty\text{?}\)
In each following question, find a formula for a polynomial with certain properties, generate a plot that demonstrates you’ve found a function with the given specifications, and write several sentences to explain your thinking.
- A quadratic function \(q\) has zeros at \(x = −7\) and \(x = 11\) and its \(y\)-value at its vertex is \(42\text{.}\)
- A polynomial \(r\) of degree \(4\) has zeros at \(x = −3\) and \(x = 5\text{,}\) both of multiplicity \(2\text{,}\) and the function has a \(y\)-intercept at the point \((0, 28)\text{.}\)
- A polynomial \(f\) has degree \(11\) and the following zeros: zeros of multiplicity \(1\) at \(x = −3\) and \(x = 5\text{,}\) zeros of multiplicity \(2\) at \(x = −2\) and \(x = 3\text{,}\) and a zero of multiplicity \(3\) at \(x = 1\text{.}\) In addition, \(\lim_{x \to \infty} f(x) = -\infty\text{.}\)
- A polynomial \(g\) has its graph given in Figure \(\PageIndex{16}\) below. Determine a possible formula for \(g(x)\) where the polynomial you find has the lowest possible degree to match the graph. What is the degree of the function you find?
Like we have worked to understand families of functions that involve parameters such as \(p(t) = a\cos(k(t-b)) + c\) and \(F(t) = a + be^{-kt}\text{,}\) we are often interested in polynomials that involve one or more parameters and understanding how those parameters affect the function's behavior.
For example, let \(a \gt 0\) be a positive constant, and consider \(p(x) = x^3 - a^2x\text{.}\)
- What is the degree of \(p\text{?}\)
- What is the long-term behavior of \(p\text{?}\) State your responses using limit notation.
- In terms of the constant \(a\text{,}\) what are the zeros of \(p\text{?}\)
- Construct a carefully labeled sign chart for \(p\text{.}\)
- How does changing the value of \(a\) affect the graph of \(p\text{?}\)
<1> We can actually say even more: if we allow the zeros to be complex numbers, then every degree \(n\) polynomial has exactly \(n\) zeros, provided we count zeros according to their multiplicity. For example, the polynomial \(p(x) = (x-1)^2 = x^2 - 2x + 1\) because it has a zero of multiplicity two at \(x = 1\text{.}\)
<2> We skip using
e
as one of the constants since
Desmos
reserves
e
as the Euler constant.