Understanding Polynomial Functions: Videos & Practice Problems

Polynomial functions are expressions with positive whole number exponents, written in standard form with descending powers. The degree indicates the highest power, while the leading coefficient affects end behavior: a positive coefficient results in rising ends, and a negative one leads to falling ends. The maximum number of turning points is determined by the degree, specifically $n-1$. Understanding zeros and their multiplicity helps predict graph behavior at intercepts, where even multiplicities touch the x-axis and odd multiplicities cross it.

Introduction to Polynomial Functions

Polynomial functions are a broad category of mathematical expressions characterized by their use of positive whole number exponents. A polynomial function can be expressed in the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n \) represents the leading coefficient, \( n \) is the degree of the polynomial, and the terms are arranged in descending order of power. For example, in the polynomial \( f(x) = 5x^3 - x^2 - 6x + 4 \), the leading coefficient is 5, and the degree is 3.

To determine if a given expression is a polynomial function, it is essential to check that all exponents are positive whole numbers. For instance, \( f(x) = 2x^{1/2} + 3 \) is not a polynomial function due to the fractional exponent. However, \( f(x) = -\frac{2}{3}x^4 + 4 \) is valid since the exponent is a positive whole number, even though it has a fractional coefficient.

The graphs of polynomial functions exhibit two key characteristics: they are smooth and continuous. This means that the graph does not have any corners or breaks. For example, a quadratic function, which is a specific type of polynomial function, maintains this smooth and continuous nature. In contrast, graphs that display sharp corners or breaks do not represent polynomial functions.

Additionally, the domain of polynomial functions is always all real numbers, extending from negative infinity to positive infinity. This property is consistent across all polynomial functions, including quadratics.

In summary, understanding the structure and characteristics of polynomial functions is crucial for analyzing their behavior and graphing them effectively. Recognizing the importance of positive whole number exponents, standard form, and the smooth, continuous nature of their graphs will aid in mastering this fundamental concept in mathematics.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

• A. Polynomial with $n=3,a_{n}=4$
• B. Polynomial with $n=4,a_{n}=3$
• C. Polynomial with $n=-1,a_{n}=\frac12$
• D. Not a polynomial function.

Answer: (D)

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=2+x$

• A. Polynomial with $n=1,a_{n}=2$
• B. Polynomial with $n=0,a_{n}=1$
• C. Polynomial with $n=1,a_{n}=1$
• D. Not a polynomial function.

Answer: (C)

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

• A. Polynomial with $n=3,a_{n}=2$
• B. Polynomial with $n=2,a_{n}=3$
• C. Polynomial with $n=2,a_{n}=2$
• D. Not a polynomial function.

Answer: (B)

End Behavior of Polynomial Functions

Understanding the end behavior of polynomial functions is crucial for graphing and analyzing their characteristics. The end behavior describes how the graph behaves as the input values approach positive or negative infinity. This behavior can be determined by examining the leading term of the polynomial when it is expressed in standard form.

The leading term consists of two key components: the leading coefficient and the degree of the polynomial. The leading coefficient indicates the direction of the graph on the right side. If the leading coefficient is positive, the graph will rise towards positive infinity as x approaches positive infinity. Conversely, if the leading coefficient is negative, the graph will fall towards negative infinity.

For example, if we have a polynomial where the leading coefficient is negative, such as in the function \( f(x) = -4x^6 + x^3 + 2 \), we can conclude that the right side of the graph will fall. Since the degree of this polynomial is 6 (an even number), both ends of the graph will exhibit the same behavior, meaning the left side will also fall.

On the other hand, the degree of the polynomial also plays a significant role in determining the end behavior. If the degree is even, both ends of the graph will behave similarly (either both rising or both falling). If the degree is odd, the ends will behave oppositely; one side will rise while the other falls. This can be remembered with the phrase "odd, opposite." For instance, in the polynomial \( f(x) = 2x^3 + x \), the leading coefficient is positive, indicating that the right side rises, while the degree is 3 (odd), suggesting that the left side must fall.

In summary, to analyze the end behavior of a polynomial function, focus on the leading term. Determine the sign of the leading coefficient to understand the behavior on the right side, and assess the degree to establish whether the ends will behave the same or oppositely. This approach simplifies the process of sketching the graph and predicting its behavior at the extremes.

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$

• A. Right side rises; Ends are same
• B. Right side rises; Ends are opposite
• C. Right side falls; Ends are same
• D. Right side falls; Ends are opposite

Answer: (B)

Match the given polynomial function to its graph based on end behavior. $f\left(x\right)=-2x^3+x^2+1$

• A.
• B.
• C.

Answer: (B)

Finding Zeros & Their Multiplicity

When working with polynomial functions, finding the x-intercepts involves setting the function \( f(x) \) equal to zero. This process is similar to what we do with quadratic functions, but polynomial functions can have three or more x-intercepts. To find these intercepts, we factor the polynomial and set each factor equal to zero. The concept of multiplicity is crucial here, as it indicates how many times a particular factor appears in the polynomial.

For example, consider the polynomial \( f(x) = 2x(x - 3)^2(x + 4)^3 \). To find the x-intercepts, we set each factor to zero:

• From \( 2x = 0 \), we find \( x = 0 \).
• From \( (x - 3)^2 = 0 \), we find \( x = 3 \).
• From \( (x + 4)^3 = 0 \), we find \( x = -4 \).

This gives us three x-intercepts: \( x = 0 \), \( x = 3 \), and \( x = -4 \). Next, we determine the multiplicity of each zero:

• The zero \( x = 0 \) comes from the factor \( 2x \), which has a multiplicity of 1 (occurs once).
• The zero \( x = 3 \) comes from the factor \( (x - 3)^2 \), which has a multiplicity of 2 (occurs twice).
• The zero \( x = -4 \) comes from the factor \( (x + 4)^3 \), which has a multiplicity of 3 (occurs three times).

Understanding multiplicity is essential for predicting the behavior of the graph at each x-intercept. If the multiplicity is even, the graph will touch the x-axis and bounce back, while an odd multiplicity indicates that the graph will cross the x-axis. In our example:

• At \( x = 0 \) (multiplicity 1), the graph crosses the x-axis.
• At \( x = 3 \) (multiplicity 2), the graph touches the x-axis and bounces back.
• At \( x = -4 \) (multiplicity 3), the graph crosses the x-axis again.

In summary, finding the x-intercepts of polynomial functions involves factoring the polynomial, setting each factor to zero, and analyzing the multiplicity of each zero to understand the graph's behavior at those points.

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$

• A. Touch at $x=0$, Cross at $x=-3$
• B. Touch at $x=0$, Touch at $x=3$
• C. Touch at $x=1,$ Cross at $x=-3$
• D. Touch at $x=-1$, Cross at $x=0$

Answer: (B)

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=x^2\left(x-1\right)^3\left(2x+6\right)$

• A. Cross at $x=0,$ Cross at $x=1,$ Cross at $x=3$
• B. Touch at $x=0,$ Cross at $x=-1,$ Cross at $x=3$
• C. Cross at $x=0,$ Touch at $x=1,$ Touch at $x=-3$
• D. Touch at $x=0,$ Cross at $x=1,$ Cross at $x=-3$

Answer: (D)

Maximum Turning Points of a Polynomial Function

Understanding the behavior of polynomial functions involves analyzing various elements, including end behavior, zeros, and turning points. The end behavior describes how the graph behaves as the variable x approaches negative or positive infinity, while the zeros indicate where the graph intersects the x-axis. At these zeros, the graph can either cross the x-axis or touch it and bounce back, which is crucial for sketching the graph accurately.

Turning points are where the graph changes direction, transitioning from increasing to decreasing or vice versa. The maximum number of turning points in a polynomial function is determined by the degree of the polynomial, denoted as n. The formula to find the maximum number of turning points is:

Maximum Turning Points = n - 1

For example, consider the polynomial function f(x) = 6x⁴ + 2x. The degree here is 4, so the maximum number of turning points is:

4 - 1 = 3

This means the graph can have up to three turning points, although it may have fewer. Each turning point can be classified as either a local maximum or a local minimum. A local maximum occurs at a point where the graph is higher than the points on either side, resembling the top of a hill. Conversely, a local minimum occurs at a point where the graph is lower than the points on either side, resembling the bottom of a valley.

To further illustrate this, consider another example: f(x) = x² - 1. The degree of this polynomial is 2, leading to a maximum of:

2 - 1 = 1

This aligns with the known shape of a quadratic function, which has one turning point. Lastly, for the polynomial f(x) = -x² + 5x³ - 6x, the degree is 3, resulting in a maximum of:

3 - 1 = 2

Knowing the maximum number of turning points is essential for verifying the accuracy of a graphed polynomial function. This understanding aids in predicting the graph's behavior and ensuring it aligns with the mathematical properties of polynomials.

Determine the maximum number of turning points for the given polynomial function. $f\left(x\right)=6x^4+2x$

• A. 1
• B. 2
• C. 3
• D. 4

Answer: (C)

Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f\left(x\right)=x^3+1$

• A.
• B.
• C.

Answer: (C)

The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. $4x^5$

• A. Right side rises; Ends are opposite & 4 maximum turning points
• B. Right side rises; Ends are opposite & 5 maximum turning points
• C. Right side rises; Ends are the same & 4 maximum turning points
• D. Right side falls; Ends are opposite & 4 maximum turning points

Answer: (A)

Here’s what students ask on this topic:

What is a polynomial function and how is it different from a polynomial expression?

A polynomial function is a mathematical expression where a polynomial is set equal to a function, typically denoted as f(x). It consists of terms with positive whole number exponents and is written in standard form with descending powers. The key characteristics of a polynomial function include its leading coefficient and degree. In contrast, a polynomial expression is simply a sum of terms with positive whole number exponents, without being equated to a function. The main difference is that a polynomial function represents a relationship between variables, allowing it to be graphed, whereas a polynomial expression is just a collection of terms.

How do you determine the end behavior of a polynomial function?

The end behavior of a polynomial function is determined by its leading term, which is the term with the highest degree when the polynomial is written in standard form. The leading coefficient and the degree of the polynomial are crucial. If the leading coefficient is positive, the right side of the graph will rise; if negative, it will fall. The degree of the polynomial also plays a role: if it is even, both ends of the graph will behave the same (either both rising or both falling); if it is odd, the ends will behave oppositely. This behavior helps predict how the graph extends towards positive and negative infinity.

What is the significance of the degree of a polynomial function?

The degree of a polynomial function is the highest power of the variable in the polynomial. It is significant because it determines the maximum number of turning points the graph can have, which is one less than the degree. The degree also influences the end behavior of the graph: an even degree means the ends of the graph will behave the same, while an odd degree means they will behave oppositely. Additionally, the degree indicates the maximum number of x-intercepts the graph can have, as it represents the number of times the function can potentially cross the x-axis.

How do you find the x-intercepts of a polynomial function?

To find the x-intercepts of a polynomial function, set the function equal to zero and solve for x. This often involves factoring the polynomial into its component factors. Each factor set to zero gives a potential x-intercept. The multiplicity of each zero, which is the number of times a factor occurs, affects the graph's behavior at that intercept. If the multiplicity is even, the graph will touch and bounce off the x-axis at that point; if it is odd, the graph will cross the x-axis. This method helps identify all points where the graph intersects the x-axis.

What are turning points in the graph of a polynomial function?

Turning points in the graph of a polynomial function are points where the graph changes direction from increasing to decreasing or vice versa. These points are significant because they represent local maxima or minima. The maximum number of turning points a polynomial graph can have is one less than the degree of the polynomial. Understanding turning points is crucial for sketching the graph accurately, as they indicate where the graph will change its slope. They also help verify the correctness of a graph by ensuring the number of turning points aligns with the polynomial's degree.