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Module 9 (Polynomial Functions) – Sections 3.2, 3.6 MAT 111

A. Polynomial Functions

We have already studied several polynomial functions.

Today we expand our knowledge of these functions, as well as higher degree polynomial functions. Notice the domain of a polynomial function is ________________________________. Furthermore, the graph of a polynomial function is always smooth (no sharp corners or points) and continuous (no breaks or holes).

Polynomial: Not Polynomials:

Name Degree Example Graph

Constant Function

Linear Function

Quadratic Function

Cubic Function

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Polynomial Function

A polynomial function is a function defined by

𝒇(𝒙) = 𝒂𝒏𝒙𝒏 + 𝒂𝒏−𝟏𝒙𝒏−𝟏 + ⋯ 𝒂𝟐𝒙𝟐 + 𝒂𝟏𝒙 + 𝒂𝟎 where the exponents 𝒏, 𝒏 − 𝟏, … , 𝟐, 𝟏, 𝟎 are whole numbers and the coefficients 𝒂𝒏, 𝒂𝒏−𝟏, … , 𝒂𝟐, 𝒂𝟏, 𝒂𝟎 are real numbers (𝒂𝒏 ≠ 𝟎).

The degree of the polynomial function is 𝒏, which is the largest power of x that appears.

The leading coefficient is 𝒂𝒏

The leading term is 𝒂𝒏𝒙𝒏

Ex1. Identify polynomial functions. If it is a polynomial function, state the degree and identify

the leading term. If not, explain why not.

Polynomial Function? Leading Term Degree

a. 𝒇(𝒙) = 𝟗

b. 𝒈(𝒙) = 𝟓𝒙𝟐 – 𝟑𝒙𝟒 + 𝟖

c. 𝒉(𝒙) = 𝟕(𝒙 + 𝟐)(𝒙 − 𝟏)𝟐

d. 𝒗(𝒙) = √𝒙 + 𝟒𝒙

e. 𝒘(𝒙) =𝟐

𝒙−

𝟖

𝒙𝟓 + 𝟔𝒙𝟑

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B. End Behavior of Polynomial Functions

End Behavior

The end behavior of a function is the general direction the function follows

as 𝑥 approaches ∞ or −∞.

The end behavior of a polynomial function 𝑓(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0

is determined by its leading term 𝒂𝒏𝒙𝒏 (𝑛 > 0).

The degree 𝒏 is even The degree 𝒏 is odd

𝒂𝒏 is positive 𝒂𝒏 is negative 𝒂𝒏 is positive 𝒂𝒏 is negative

Think 𝒚 = 𝒙𝟐

Think 𝒚 = −𝒙𝟐

Think 𝒚 = 𝒙𝟑

Think 𝒚 = −𝒙𝟑

Ex2. Determine the end behavior of the following polynomial functions.

Example Leading Term End Behavior

a. 𝒇(𝒙) = −𝟒𝒙𝟓 + 𝟔𝒙𝟒 − 𝟐𝒙

b. 𝒈(𝒙) = 𝟎. 𝟓𝒙(𝒙 − 𝟒)𝟐(𝒙 + 𝟏)𝟑

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C. Zeros and the y-intercept of Polynomial Functions

Zeros and the y-intercept

Consider a polynomial function defined by 𝑦 = 𝑓(𝑥). The y-intercept is 𝒇(𝟎).

The zeros (x-intercepts, roots) are the 𝑥-values in the domain for which 𝒇(𝒙) = 𝟎.

A polynomial function of degree 𝑛 can have at most 𝑛 real zeros.

Let (𝒙 − 𝒄)𝒌 be a factor of the polynomial where 𝒙 = 𝒄 is a real zero of the polynomial

with multiplicity 𝒌. If 𝑥 = 𝑐 is a real zero of even multiplicity 𝑘, then the graph touches the 𝑥-axis at 𝑐. If 𝑥 = 𝑐 is a real zero of odd multiplicity 𝑘, then the graph crosses the 𝑥-axis at 𝑐.

𝒇(𝒙) = (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐

Ex3. Given 𝒇(𝒙) = −𝟑𝒙𝟐(𝒙𝟐 − 𝟒)(𝒙𝟐 + 𝟗)(𝒙 − 𝟕)𝟒, identify the real zeros of the function and

whether the graph touches or crosses the x-axis at each zero.

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D. Turning Points of Polynomial Functions

Turning Points

The local maximum and minimum points on the graphs of polynomials are also known as turning points. (Keep in mind that a zero of even multiplicity will also be a local maximum or minimum point).

For a polynomial of degree n, the graph will have at most 𝒏 − 𝟏 turning points.

Ex4. Use the graph of the given function to answer the following questions.

a. How many turning points does the function have?

b. What is a possible degree of the function? c. What is the sign of the coefficient of the leading term of the function? d. List the zeros and whether each zero must have even or odd multiplicity. e. Write a possible equation for the given polynomial.

−5 0 4 8

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E. Graph Polynomial Functions

Ex5. 𝒇(𝒙) = −𝟓𝒙𝟒 − 𝟓𝒙𝟑 + 𝟑𝟎𝒙𝟐 Leading Term: _______________________

Degree: _______________________________ Max # of turning points: ___________ End-behavior: Y-intercept: __________________________ *********************************************************************************************** Ex6. 𝒇(𝒙) = 𝟐(𝒙 + 𝟏)𝟒(𝒙 − 𝟏)𝟕(𝒙 − 𝟓)𝟐 Leading Term: _______________________ Degree: _______________________________

Max # of turning points: ___________ End-behavior: Y-intercept: __________________________

Zeros Multiplicity Touch or Cross

Zeros Multiplicity Touch or Cross

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F. Polynomial Inequalities

𝒇(𝒙) = (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐

Ex7. (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐 > 𝟎

Ex8. (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐 ≥ 𝟎

Ex9. (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐 < 𝟎

Ex10. (𝒙 + 𝟑)(𝒙 − 𝟏)𝟐 ≤ 𝟎

Solving Polynomial Inequalities

STEP 1 Write the inequality so that the polynomial expression 𝑓(𝑥) is on the left side and zero is on the right side.

STEP 2 Determine the real zeros (x-intercepts) of the graph of 𝑓(𝑥).

STEP 3 Use the real zeros found in STEP 2 to divide the real number line (x-axis) into intervals.

STEP 4

Select a number in each interval, evaluate 𝑓(𝑥) at the number, and determine whether 𝑓(𝑥) is positive or negative. If 𝑓(𝑥) is positive, all values of x in the interval are positive and the graph is above the x-axis. If 𝑓(𝑥) is negative, all values of x in the interval are negative and the graph is below the x-axis.

STEP 5 Write your answer to the inequality using interval notation.

−∞ ∞ -3 1

−∞ ∞ -3 1

−∞ ∞ -3 1

−∞ ∞ -3 1

−∞ ∞ -3 1

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Ex11. 𝒙𝟒 < 𝟗𝒙𝟐

****************************************************************************************************

Ex12. −𝒙(𝒙 + 𝟏)𝟐(𝒙 − 𝟐)𝟑(𝒙 − 𝟒) ≥ 𝟎

−∞ ∞

−∞ ∞