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A polynomial of degree n is a function of the form
The coefficient of the term with the highest degree (greatest power of x) is called the leading coeficient and cannot equal 0.
A polynomial can have any non-negative degree. In other words, the polynomial's degree n must be ≥ 0.
a) is a polynomial of degree 4 with leading coefficient 3.
b) is not a polynomial because one of terms has a negative degree.
c) is a polynomial of degree 4 with a leading coefficient of -3. Note that the degree of the polynomial is the greatest power to which x is raised and the leading coefficient is the coefficient of that power regardless of the order in which the terms are written.
A polynomial of degree 0, , where k is a constant, is called a constant function.
A polynomial of degree 1, , is called a linear function. Polynomials of degrees 2, 3, and 4 are known as quadratic, cubic, and quartic functions, respectively.
-> see also: Factoring & Expanding
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 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. The following phrases are equivalent:
- "find the zeroes of f(x)"
- "find the roots of f(x)"
- "find all the x-intercepts of the graph of f(x)"
- "find all the solutions to f(x)=0"
To solve such a problem, the polynomial needs to be in factored form. Since the product of 0 and anything is 0, the roots are found by setting each term in factored form equal to 0.
Problem: Find the roots of .
Solution: We must first put the equation into factored form.
Pulling out an x, we get . Since factors to , we have .
Then, to find the roots, we solve the equation .
Solving , , and , we find that the roots are , , and .
Problem: Find all roots of and give their multiplicities.
Solution: We can write the function as and find the roots by solving .
Setting each term individually equal to zero, we get , , , , and . Thus, the two unique roots are and , which have multiplicities of 2 and 3, respectively.
Knowing the multiplicity of a root is useful in determing features of a polynomial's graph, as will be discussed in the next section.
-> see also: Manipulating Graphs
One can determine many features of the graph of a polynomial using only precalculus. We are interested in being able to create graphs of polynomials that show their end behavior, y-intercept, x-intercepts, and behavior at roots. Let's look at each of these features individually.
The end behavior of any given polynomial is identical to the end behavior of its parent polynomial, . 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. Below are graphs of the functions y=x, y=x2, y=x3, y=x4, y=x5, and y=x6. What do you notice about the end behavior in each case?
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, . 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 end behavior of a polynomial also depends on the sign of its leading coefficient. The cases examined thus far have all had positive leading coefficients (in fact, the leading coefficient has been 1 in each case). But what if the leading coefficient is negative? The figures below show graphs of y=-x, y=-x2.
As can be seen in the graphs above, the negative leading coefficient flips the graphs over the x-axis (see more: manipulating graphs), and, consequently, flips the end behavior. Thus, for any polynomial with a negative leading coefficient, . Furthermore, if n is odd, while if n is even.
Problem: Give the end behavior for each of the following polynomials.
a) The polynomial is of degree 7, which is odd. Its leading coefficient is 12, which is positive. (If you don't see why, notice that if you multiply the function out, you get ). Thus, the polynomial has the same end behavior as ; namely, and .
b) 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. 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. Therefore, and .
The y-intercept of a polynomial is where the graph crosses the y-axis. It can be easily found be evaluating the function at the point x=0.
Problem: Find the y-intercept of .
Solution: Simply evaluate to find the y-intercept is .
Recall that x-intercepts are another name for the roots or zeroes of a polynomial. Thus, x-intercepts on a graph are found exactly as described above in the section Finding Roots.
Problem: Mark the x-intercepts of the polynomial on a graph.
Solution First, factor the polynomial completely to find and set this equal to zero: . Solving the equation, we find the roots are . The diagram below shows these points marked on the x-axis.
Behavior at Roots
We know that the graph of the polynomial touches the x-axis at each of its roots. The next logical thing to wonder is what it does at the points immediately around these x-intercepts. There are two possibilities: either the graph approaches the root from one side of the x-axis, passes through the axis, and continues on the other side; or else it approaches the root from one side, touches the axis, but then bounces off and continues on the same side of the axis from which it approached. These two cases are illustrated below.
"Passing through" the x-axis: "Bouncing off" the x-axis:
How does one know which behavior will occur at a root? It depends on the multiplicity of the root. Consider the graphs of y=x2 and y=x3, below.
Both have the same root, x=0. But the graph of y=x2 bounces off the axis at the root, while the graph of y=x3 passes through. What's the difference between these two roots? Their multiplicities. If a root has an even multiplicity, the graph will bounce off the axis, as with the graph of y=x2. If a root has an odd multiplicity, the graph will pass through the axis, as with the graph of y=x3.
Problem: For each of the polynomials below, determine the roots and whether the graph passes through or bounces off the axis at each point.
a) Setting the function equal to 0, one finds the roots are x=1 and x=-2, which have multiplicities of 1 and 2, respectively. Thus, the graph will pass through the axis at the root x=1 and bounce off the axis at the root x=-2.
b) First, factor the polynomial: . Setting the function equal to 0, one finds the roots are x=-2 and x=-3, which have multiplicities of 3 and 2, respectively. Thus, the graph will bounce off the axis at the root x=-2 and pass through the axis at the root x=-3.
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.
PRACTICE PROBLEMS: (top | answers | solutions)
Graph the following polynomials:
This page last updated 20 June, 2008 11:58 PM