PROJECT 3
Roots and Graphs of Polynomials
Roots and their Multiplicity
We begin by considering the polynomial
Recall that if is a factor of p, then c is a root (zero) of p and that the degree, n, of p tell us the maximum number of roots. If we use other theorems, which we will neither state nor prove here, we can also find out that every polynomial can be factored as
, where each is a complex number.
From experience, we know that each of the numbers are not necessarily all different. To illustrate this point, let us look at the polynomial
.
We can factor f as:
.
If a factor occurs m times in the factorization, then x is a root of multiplicity m of f(x). In the preceding example, 1 is a root of multiplicity 2 and 1 is a root of multiplicity 1.
Example1:
Find the roots of the polynomial , and state the multiplicity of each.
Solution:
We see from the factored form that the roots are 1, 4 and 1. From the power of their corresponding factors, we see that 1 is a root of multiplicity 1, 4 is a root of multiplicity 3, and 1 is a root of multiplicity 2.
The multiplicity of a root has an effect on the shape of the graph near the root. A root with multiplicity 2 will have behavior similar to , a root of multiplicity 3 will have behavior similar to , a root of multiplicity 4 will have behavior like , etc. (only close to the root).
Descartes’ Rule of Signs
We can use Descartes’s rule of signs to give us a bit more information about our polynomial. We will assume the p(x) is a polynomial with real coefficients and nonzero constant term. The rule follows:
- The number of positive real roots of p(x) is equal to the number of variations of sign in p(x) or is less than that number by an even integer.
- The number of negative real roots of p(x) is equal to the number of variations of sign in p(x) or is less than that number by an even integer.
Example 2:
Discuss the number of positive and negative roots of the polynomial:
Solution:
We note that three are 3 variations of sign I f(x), hence the function has 3 or 1 positive real roots. Since has two variations of sign, we can conclude that f(x) has 2 or no negative real roots. Since the degree of f is 5, we can conclude that:
f / has / 3 / positive real roots, / 2 / negative real roots, and / 0 / complex roots / orf / has / 3 / positive real roots, / 0 / negative real roots, and / 2 / complex roots / or
f / has / 1 / positive real roots, / 2 / negative real roots, and / 2 / complex roots / or
f / has / 1 / positive real roots, / 0 / negative real roots, and / 4 / complex roots
The Rational Root Test
Although we can find all the roots of a quadratic polynomial using the quadratic formula and there are analogous more complicated formulas for finding the roots of cubics and quartics, it is not possible to find all the roots of any polynomial of degree 5 or greater. We can though find all the rational roots of any polynomial with integer coefficients. To do this we use the Rational Root Test. It is stated as follows:
If a rational number (in lowest terms) is a root of the polynomial where the coefficients are integers with and , then r is a factor of the constant term and s is a factor of the leading coefficient .
Example 3:
Find all the possible rational roots of .
Solution:
The factors of the constant term 20 are 1, 2, 4, 5, 10, and 20. The factors of the leading coefficient 3 are 1 and 3. If we form all possible ratios of these two sets of numbers, we will have all the possible rational roots. The possible roots are
To determine which are actual roots, you would need to plug them into the polynomial.
Turns of Polynomials
The degree of a polynomial also determines the shape of the graph, as shown in Figure 1 below. These graphs correspond to a positive coefficient for ; a negative coefficient flips the graph over. Notice that the graph of the quadratic “turns around” once, the cubic “turns around” twice, and the quartic (fourth degree) “turns around” three times. An degree polynomial “turns around” at most times (where n is a positive integer), but there may be fewer turns.
Figure 1: Graphs of typical polynomials of degree n
Long-Run or End Behavior of Polynomial Functions
As gets very large (in other words as x approaches or as x approaches ), the graph of a polynomial function closely resembles the graph of its highest degree term. More specifically:
if p(x) is an odd-degree polynomial function then
if the leading coefficient is positive,
then as and as ;
if the leading coefficient is negative,
then as and as .
if p(x) is an even-degree polynomial function then
if the leading coefficient is positive,
then as and as ;
if the leading coefficient is negative,
then as and as .
Finding the Formula for a Polynomial from its Graph
Given the graph of a polynomial, it is often possible to find a formula for a function that will have the same graphical behavior as the polynomial. To do this we use what we know about basic polynomial graphs and specific information from the given graph such as roots and other specific points.
Example 3:
Find possible formulas for the polynomials whose graphs are in Figure 2
Solution:
(i.)This graph appears to be a parabola, turned upside down and moved up by 4, so
The minus sign turns the parabola upside down and the +4 moves it up by 4. You should notice that this formula does five the correct x-intercepts since has solutions .
You can also solve this problem by looking at the x-intercepts first, which tell you f(x) must have factors of and , so where k is some constant. To find k, use the fact that the graph has a y-intercept of 4, so , giving
so . Therefor , which multiplies out to . Note that also fits the requirements, but its “shoulders” are sharper. There are many possible answer to these questions.
(ii.)This looks like a cubic with factors , , and , one for each intercept:
.
Since the y-intercept is 12,
so , and
.
(iii.)This also looks like a cubic with zeros at and . Notice that at the graph of h(x) touches the x-axis but does not cross it, whereas at the graph crosses the x-axis. We say that is a double zero or root with multiplicity 2, but that is a single zero or root with multiplicity 1.
To find a formula for h(x), first imagine the graph of h(x) to be slightly lower down, so that the graph has one x-intercept near and two near , say at and . Then
.
Now move the graph back to its original position. The zeros at and move toward giving
.
Thus the double zero leads to a repeated factor, . Notice that when , the factor is positive, and when , is still positive. This reflects the fact that h(x) does not change sign near . Compare this with the behavior near the single zero, where h does change sign.
You cannot find k, as no coordinates are given for points off of the x-axis. Inserting any positive value of k will stretch the graph but not change the zeros and therefore will still work.
Problems for Roots and Graphs of Polynomials
In Contemporary Precalculus: A Graphing Approach by Thomas W. Hungerford complete problems 1-12, 14, 15, 44-47 in section 4.3 page 239.
Also complete the following problems:
1.Assume that each of the graphs below is a polynomial. For each graph:
a.What is the minimum possible degree of the polynomial?
b.Is the leading coefficient of the polynomial positive or negative?
2.Describe the end or long-run behavior of each of the functions below.
a.
b.
c.
d.
e.
3.Use the graph of below to determine the factored form of .
4.Find possible formulas for the following polynomials given:
a.f is a third degree polynomial with f(3) = 0, f(1) = 0, f(4) = 0, and f(2) = 5.
b.g is a fourth degree polynomial, g has a “double zero” at x = 3, g (5) = 0, g(1) = 0, and g (0) = 3.
5.Find the real zeros of the following polynomials:
a.
b.
c.
d.
e.
f.
6.Suppose you wish to pack a cardboard box inside a wooden crate. In order to have room for the packing materials, you need to leave a 0.5-ft space around the front, back, and sides of the box and a 1-ft space around the top and bottom of the box. If the cardboard box is x feet long, (x + 2) feet wide, and (x 1) feet deep, find a formula in terms of x for the amount of packing material needed.
7.Find a possible formula for each of the polynomial functions whose graphs are shown below.
8.If and , find all x for which . (Note : When , then .)
9.For each of the following polynomial functions, state the degree of the polynomial, the number of terms in the polynomial, and describe its long-run or end behavior.
a.
b.
c.
10.Suppose f is a polynomial function of degree n, where n is a positive even integer. For each of the following statements, write true is the statement is always true, false otherwise. If the statement is false, give an example that illustrates why it is false.
a.f is even.
b.f has an inverse.
c.f cannot be an odd function.
d.If as , then as .
11.Suppose the following statements about f(x) are true:
- f(x) is a polynomial function
- f(x) = 0 at exactly four different values of x
- as
For each of the following statements, write true if the statement must be true, never true is the statement is never true, or neither if it is sometimes true and sometimes not true.
a.f(x) is odd.
b.f(x) is even.
c.f(x) is a fourth degree polynomial.
d.f(x) is a fifth degree polynomial.
e. as
f.f(x) is invertible.
12.For the polynomial graphed below, find an equation that represents is.
13.Find all the rational roots for the polynomial . (Hint: The Rational Root Test can only be used on polynomials with non-zero constant terms. You will need to factor the polynomial to find one that has a non-zero constant term.)
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