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Study Guides > MTH 163, Precalculus

Identify the degree and leading coefficient of polynomial functions

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

A General Note: Terminology of Polynomial Functions

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0. Figure 6

We often rearrange polynomials so that the powers are descending.

When a polynomial is written in this way, we say that it is in general form.

How To: Given a polynomial function, identify the degree and leading coefficient.

  1. Find the highest power of x to determine the degree function.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.

Example 5: Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

[latex]\begin{cases} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\ g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\ h\left(p\right)=6p-{p}^{3}-2\end{cases}\\[/latex]

Solution

For the function [latex]f\left(x\right)\\[/latex], the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}\\[/latex]. The leading coefficient is the coefficient of that term, –4.

For the function [latex]g\left(t\right)\\[/latex], the highest power of t is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}\\[/latex]. The leading coefficient is the coefficient of that term, 5.

For the function [latex]h\left(p\right)\\[/latex], the highest power of p is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}\\[/latex]; the leading coefficient is the coefficient of that term, –1.

Try It 3

Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\left(x\right)=4{x}^{2}-{x}^{6}+2x - 6\\[/latex].

Solution

Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.

Polynomial Function Leading Term Graph of Polynomial Function
[latex]f\left(x\right)=5{x}^{4}+2{x}^{3}-x - 4\\[/latex] [latex]5{x}^{4}\\[/latex] Graph of f(x)=5x^4+2x^3-x-4.
[latex]f\left(x\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}\\[/latex] [latex]-2{x}^{6}\\[/latex] Graph of f(x)=-2x^6-x^5+3x^4+x^3.
[latex]f\left(x\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1\\[/latex] [latex]3{x}^{5}\\[/latex] Graph of f(x)=3x^5-4x^4+2x^2+1.
[latex]f\left(x\right)=-6{x}^{3}+7{x}^{2}+3x+1\\[/latex] [latex]-6{x}^{3}\\[/latex] Graph of f(x)=-6x^3+7x^2+3x+1.

Example 6: Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.
Graph of an odd-degree polynomial. Figure 7

Solution

As the input values x get very large, the output values [latex]f\left(x\right)\\[/latex] increase without bound. As the input values x get very small, the output values [latex]f\left(x\right)\\[/latex] decrease without bound. We can describe the end behavior symbolically by writing

[latex]\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to \infty \end{cases}\\[/latex]

In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Try It 4

Describe the end behavior, and determine a possible degree of the polynomial function in Figure 9.
Graph of an even-degree polynomial. Figure 9
Solution

Example 7: Identifying End Behavior and Degree of a Polynomial Function

Given the function [latex]f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\[/latex], express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Solution

Obtain the general form by expanding the given expression for [latex]f\left(x\right)\\[/latex].

[latex]\begin{cases} f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\ \hfill =-3{x}^{2}\left({x}^{2}+3x - 4\right)\\ \hfill=-3{x}^{4}-9{x}^{3}+12{x}^{2}\end{cases}\\[/latex]

The general form is [latex]f\left(x\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\[/latex]. The leading term is [latex]-3{x}^{4}\\[/latex]; therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

[latex]\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to -\infty \end{cases}\\[/latex]

Try It 5

Given the function [latex]f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)\\[/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Solution

Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the "middle" of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

Figure 10

We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept [latex]\left(0,{a}_{0}\right)\\[/latex]. The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept. 

A General Note: Intercepts and Turning Points of Polynomial Functions

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.

How To: Given a polynomial function, determine the intercepts.

  1. Determine the y-intercept by setting [latex]x=0\\[/latex] and finding the corresponding output value.
  2. Determine the x-intercepts by solving for the input values that yield an output value of zero.

Example 8: Determining the Intercepts of a Polynomial Function

Given the polynomial function [latex]f\left(x\right)=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)\\[/latex], written in factored form for your convenience, determine the y- and x-intercepts.

Solution

The y-intercept occurs when the input is zero so substitute 0 for x.

[latex]\begin{cases}f\left(0\right)=\left(0 - 2\right)\left(0+1\right)\left(0 - 4\right)\hfill \\ \text{ }=\left(-2\right)\left(1\right)\left(-4\right)\hfill \\ \text{ }=8\hfill \end{cases}\\[/latex]

The y-intercept is (0, 8).

The x-intercepts occur when the output is zero.

[latex]\begin{cases}\text{ }0=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)\hfill \\ x - 2=0\hfill & \hfill & \text{or}\hfill & \hfill & x+1=0\hfill & \hfill & \text{or}\hfill & \hfill & x - 4=0\hfill \\ \text{ }x=2\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=-1\hfill & \hfill & \text{or}\hfill & \hfill & x=4 \end{cases}[/latex]

The x-intercepts are [latex]\left(2,0\right),\left(-1,0\right)\\[/latex], and [latex]\left(4,0\right)\\[/latex].

We can see these intercepts on the graph of the function shown in Figure 11.
Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts. Figure 11

Example 9: Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function [latex]f\left(x\right)={x}^{4}-4{x}^{2}-45\\[/latex], determine the y- and x-intercepts.

Solution

The y-intercept occurs when the input is zero.

[latex]\begin{cases} \\ f\left(0\right)={\left(0\right)}^{4}-4{\left(0\right)}^{2}-45\hfill \hfill \\ \text{ }=-45\hfill \end{cases}\\[/latex]

The y-intercept is [latex]\left(0,-45\right)\\[/latex].

The x-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

[latex]\begin{cases}f\left(x\right)={x}^{4}-4{x}^{2}-45\hfill \\ =\left({x}^{2}-9\right)\left({x}^{2}+5\right)\hfill \\ =\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\hfill \end{cases}[/latex]
[latex]0=\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\\[/latex]
[latex]\begin{cases}x - 3=0\hfill & \text{or}\hfill & x+3=0\hfill & \text{or}\hfill & {x}^{2}+5=0\hfill \\ \text{ }x=3\hfill & \text{or}\hfill & \text{ }x=-3\hfill & \text{or}\hfill & \text{(no real solution)}\hfill \end{cases}\\[/latex]

The x-intercepts are [latex]\left(3,0\right)\\[/latex] and [latex]\left(-3,0\right)\\[/latex].

We can see these intercepts on the graph of the function shown in Figure 12. We can see that the function is even because [latex]f\left(x\right)=f\left(-x\right)\\[/latex].
Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45). Figure 12

Try It 6

Given the polynomial function [latex]f\left(x\right)=2{x}^{3}-6{x}^{2}-20x\\[/latex], determine the y- and x-intercepts.

Solution

Comparing Smooth and Continuous Graphs

The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. A polynomial function of nth degree is the product of n factors, so it will have at most n roots or zeros, or x-intercepts. The graph of the polynomial function of degree n must have at most n – 1 turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

A General Note: Intercepts and Turning Points of Polynomials

A polynomial of degree n will have, at most, n x-intercepts and n – 1 turning points.

Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for [latex]f\left(x\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}\\[/latex].

Solution

The polynomial has a degree of 10, so there are at most n x-intercepts and at most n – 1 turning points.

Try It 7

Without graphing the function, determine the maximum number of x-intercepts and turning points for [latex]f\left(x\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}\\[/latex]

Solution

Example 11: Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in the graph in Figure 13 based on its intercepts and turning points?
Graph of an even-degree polynomial. Figure 13

Solution

Graph of an even-degree polynomial that denotes the turning points and intercepts. Figure 14

The end behavior of the graph tells us this is the graph of an even-degree polynomial. 

The graph has 2 x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

Try It 8

What can we conclude about the polynomial represented by Figure 15 based on its intercepts and turning points?
Graph of an odd-degree polynomial. Figure 15
Solution

Example 12: Drawing Conclusions about a Polynomial Function from the Factors

Given the function [latex]f\left(x\right)=-4x\left(x+3\right)\left(x - 4\right)\\[/latex], determine the local behavior.

Solution

The y-intercept is found by evaluating [latex]f\left(0\right)\\[/latex].

[latex]\begin{cases}f\left(0\right)=-4\left(0\right)\left(0+3\right)\left(0 - 4\right)\hfill \hfill \\ \text{ }=0\hfill \end{cases}\\[/latex]

The y-intercept is [latex]\left(0,0\right)\\[/latex].

The x-intercepts are found by determining the zeros of the function.

[latex]\begin{cases}0=-4x\left(x+3\right)\left(x - 4\right)\\ x=0\hfill & \hfill & \text{or}\hfill & \hfill & x+3=0\hfill & \hfill & \text{or}\hfill & \hfill & x - 4=0\hfill \\ x=0\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=-3\hfill & \hfill & \text{or}\hfill & \hfill & \text{ }x=4\end{cases}\\[/latex]

The x-intercepts are [latex]\left(0,0\right),\left(-3,0\right)\\[/latex], and [latex]\left(4,0\right)\\[/latex].

The degree is 3 so the graph has at most 2 turning points.

Try It 9

Given the function [latex]f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)\\[/latex], determine the local behavior.

Solution

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