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Lesson 1: Polynomial Functions

Estimated time: 30-35 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Identify polynomial functions and determine their degree and leading coefficient
Write polynomial functions in standard form
Determine end behavior using the Leading Term Test
Find zeros (x-intercepts) of polynomial functions
Understand and identify multiplicity of zeros
Graph polynomial functions using zeros, end behavior, and multiplicity

What is a Polynomial Function?

A polynomial function is one of the most important function types in mathematics. Polynomials appear in physics, economics, engineering, and countless other applications.

Polynomial Function: A function that can be written in the form:

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

where n is a non-negative integer and the coefficients a_n, a_n-1, ..., a₀ are real numbers with a_n ≠ 0.

Example 1: Identifying Polynomial Functions

Which of the following are polynomial functions?

a) f(x) = 3x⁴ - 2x² + 7

Answer: YES - All exponents are non-negative integers.

b) g(x) = 5x³ - 2x^-1 + 1

Answer: NO - The term x^-1 has a negative exponent.

c) h(x) = 2√x + 3x - 1

Answer: NO - √x = x^1/2 is not an integer exponent.

d) k(x) = -x⁵ + 4x³ - 8x + 12

Answer: YES - All exponents are non-negative integers.

Degree and Leading Coefficient

Degree: The highest power of x in the polynomial (when written in standard form).

Leading Coefficient: The coefficient of the term with the highest degree.

Standard Form: A polynomial written with terms in descending order of degree.

Example 2: Identifying Degree and Leading Coefficient

For the polynomial f(x) = -3x⁵ + 2x³ - 7x² + 4x - 9:

Step 1: Check if it's in standard form

Yes - terms are arranged from highest to lowest degree: 5, 3, 2, 1, 0

Step 2: Identify the degree

The highest power of x is 5, so the degree is 5.

Step 3: Identify the leading coefficient

The coefficient of x⁵ is -3, so the leading coefficient is -3.

Example 3: Writing in Standard Form

Write f(x) = 5 + 3x³ - 2x + 7x⁴ - x² in standard form.

Solution:

Step 1: Identify the degree of each term

5 has degree 0
3x³ has degree 3
-2x has degree 1
7x⁴ has degree 4
-x² has degree 2

Step 2: Arrange in descending order

f(x) = 7x⁴ + 3x³ - x² - 2x + 5

Degree: 4 | Leading coefficient: 7

        Common Polynomial Names by Degree
        Degree 0: Constant function (e.g., f(x) = 5)
Degree 1: Linear function (e.g., f(x) = 2x + 3)
Degree 2: Quadratic function (e.g., f(x) = x2 - 4)
Degree 3: Cubic function (e.g., f(x) = x3 - 2x)
Degree 4: Quartic function (e.g., f(x) = x4 + x2)
Degree 5: Quintic function (e.g., f(x) = x5 - 3x3)

      

End Behavior and the Leading Term Test

The end behavior of a polynomial describes what happens to f(x) as x approaches positive infinity (+∞) or negative infinity (-∞).

Remarkably, the end behavior of any polynomial is determined entirely by its leading term (the term with the highest degree).

Leading Term Test

For f(x) = a_nxⁿ + ... + a₁x + a₀ where a_n ≠ 0:

End behavior depends on: (1) degree n and (2) sign of a_n

Even Degree, Positive Leading Coefficient

Example: f(x) = 2x⁴ - 3x² + 1

As x → -∞: f(x) → +∞

As x → +∞: f(x) → +∞

Both ends point UP

Even Degree, Negative Leading Coefficient

Example: f(x) = -x⁴ + 5x² - 2

As x → -∞: f(x) → -∞

As x → +∞: f(x) → -∞

Both ends point DOWN

Odd Degree, Positive Leading Coefficient

Example: f(x) = 3x⁵ - 2x³ + x

As x → -∞: f(x) → -∞

As x → +∞: f(x) → +∞

Left DOWN, Right UP

Odd Degree, Negative Leading Coefficient

Example: f(x) = -2x³ + 4x - 1

As x → -∞: f(x) → +∞

As x → +∞: f(x) → -∞

Left UP, Right DOWN

Example 4: Determining End Behavior

Determine the end behavior of f(x) = -4x⁶ + 3x⁴ - 2x² + 7

Step 1: Identify the leading term

Leading term: -4x⁶

Step 2: Identify degree and leading coefficient

Degree: 6 (even)

Leading coefficient: -4 (negative)

Step 3: Apply Leading Term Test

Even degree + Negative leading coefficient means:

As x → -∞, f(x) → -∞

As x → +∞, f(x) → -∞

Interpretation: Both ends of the graph point downward.

Zeros and X-Intercepts

Zero (or Root): A value of x where f(x) = 0.

X-intercept: A point where the graph crosses or touches the x-axis. Same as a zero, but written as a point (x, 0).

Finding zeros is one of the most important tasks in working with polynomials. Zeros tell us where the graph intersects the x-axis.

Example 5: Finding Zeros from Factored Form

Find all zeros of f(x) = (x - 3)(x + 2)(x - 1)

Solution:

Set f(x) = 0:

(x - 3)(x + 2)(x - 1) = 0

By the Zero Product Property, at least one factor must equal zero:

x - 3 = 0 → x = 3
x + 2 = 0 → x = -2
x - 1 = 0 → x = 1

Zeros: x = -2, x = 1, x = 3

X-intercepts: (-2, 0), (1, 0), (3, 0)

Example 6: Finding Zeros by Factoring

Find all zeros of f(x) = x³ - 4x

Step 1: Factor out the GCF

f(x) = x(x² - 4)

Step 2: Factor completely (difference of squares)

f(x) = x(x - 2)(x + 2)

Step 3: Set each factor equal to zero

x = 0
x - 2 = 0 → x = 2
x + 2 = 0 → x = -2

Zeros: x = -2, x = 0, x = 2

Important Theorem

Fundamental Theorem of Algebra: A polynomial of degree n has exactly n zeros (counting multiplicity) in the complex numbers.

For real polynomials, this means a polynomial of degree n has at most n real zeros.

Multiplicity of Zeros

Multiplicity: The number of times a particular zero appears as a factor.

If (x - c)^k is a factor of f(x), then x = c is a zero with multiplicity k.

        Graph Behavior at Zeros
        Odd multiplicity (1, 3, 5, ...): Graph CROSSES the x-axis at the zero
Even multiplicity (2, 4, 6, ...): Graph TOUCHES (bounces off) the x-axis at the zero

      

Example 7: Identifying Multiplicity

For f(x) = (x - 2)³(x + 1)²(x - 5), identify all zeros and their multiplicities.

Solution:

Zero	Multiplicity	Graph Behavior
x = 2	3 (odd)	Crosses x-axis
x = -1	2 (even)	Touches x-axis (bounces)
x = 5	1 (odd)	Crosses x-axis

Example 8: Multiplicity and Graph Behavior

Sketch the behavior of f(x) = -x²(x - 3)²(x + 2) near its zeros.

Step 1: Find zeros and multiplicities

x = 0 with multiplicity 2 (even) - graph touches/bounces
x = 3 with multiplicity 2 (even) - graph touches/bounces
x = -2 with multiplicity 1 (odd) - graph crosses

Step 2: Determine end behavior

Degree: 2 + 2 + 1 = 5 (odd)

Leading coefficient: -1 (negative)

End behavior: As x → -∞, f(x) → +∞; As x → +∞, f(x) → -∞

Step 3: Sketch behavior

The graph:

Starts from upper left (x → -∞)
Crosses x-axis at x = -2
Touches (bounces) at x = 0
Touches (bounces) at x = 3
Ends in lower right (x → +∞)

Graphing Polynomial Functions

        Steps for Graphing Polynomials
        Find the degree and leading coefficient
Determine end behavior using the Leading Term Test
Find all zeros and their multiplicities
Determine if graph crosses or touches at each zero
Find the y-intercept (evaluate f(0))
Plot points and sketch the smooth curve

      

Example 9: Complete Polynomial Graphing

Sketch the graph of f(x) = x³ - 4x² + 4x

Step 1: Factor to find zeros

f(x) = x(x² - 4x + 4) = x(x - 2)²

Step 2: Identify zeros and multiplicities

x = 0, multiplicity 1 (crosses)
x = 2, multiplicity 2 (touches/bounces)

Step 3: Determine end behavior

Degree: 3 (odd), Leading coefficient: 1 (positive)

As x → -∞, f(x) → -∞; As x → +∞, f(x) → +∞

Step 4: Find y-intercept

f(0) = 0, so y-intercept is (0, 0)

Step 5: Test a point between zeros

f(1) = 1(1 - 2)² = 1(1) = 1, so the point (1, 1) is on the graph

Step 6: Sketch

The graph:

Comes from lower left
Crosses at x = 0
Goes up to touch x = 2 and bounces back down briefly
Then continues upward to upper right

Example 10: Another Graphing Example

Analyze f(x) = -x⁴ + 4x²

Step 1: Factor

f(x) = -x²(x² - 4) = -x²(x - 2)(x + 2)

Step 2: Zeros and multiplicities

x = 0, multiplicity 2 (touches)
x = 2, multiplicity 1 (crosses)
x = -2, multiplicity 1 (crosses)

Step 3: End behavior

Degree: 4 (even), Leading coefficient: -1 (negative)

Both ends point DOWN: As x → ±∞, f(x) → -∞

Step 4: Y-intercept

f(0) = 0

Step 5: Additional points

f(1) = -(1)(1 - 2)(1 + 2) = -1(-1)(3) = 3

f(-1) = -(-1)(-1 - 2)(-1 + 2) = -(1)(-3)(1) = 3

Graph characteristics:

Starts from lower left (going down)
Crosses at x = -2
Goes up, reaches maximum near x = -1
Comes down to touch x = 0 (bounce)
Goes back up, reaches maximum near x = 1
Comes down to cross at x = 2
Continues down to lower right

Check Your Understanding

1. What is the degree and leading coefficient of f(x) = 5 - 2x³ + 4x⁵ - x?

Solution:

First, write in standard form: f(x) = 4x⁵ - 2x³ - x + 5

Degree: 5

Leading coefficient: 4

2. Determine the end behavior of f(x) = -2x⁶ + 3x⁴ - x² + 7

Solution:

Degree: 6 (even)

Leading coefficient: -2 (negative)

End behavior:

As x → -∞, f(x) → -∞

As x → +∞, f(x) → -∞

Both ends point downward.

3. Find all zeros of f(x) = x³ - 9x and state their multiplicities.

Solution:

Factor: f(x) = x(x² - 9) = x(x - 3)(x + 3)

Zeros:

x = 0 (multiplicity 1)
x = 3 (multiplicity 1)
x = -3 (multiplicity 1)

All zeros have odd multiplicity, so the graph crosses the x-axis at each zero.

4. For f(x) = (x + 1)²(x - 2)³, describe the behavior at x = -1 and x = 2.

Solution:

At x = -1: Multiplicity 2 (even) → graph touches and bounces

At x = 2: Multiplicity 3 (odd) → graph crosses

5. What is the maximum number of x-intercepts a polynomial of degree 5 can have?

Answer: 5 x-intercepts

Explanation: By the Fundamental Theorem of Algebra, a polynomial of degree n has at most n real zeros. Therefore, a degree 5 polynomial has at most 5 x-intercepts.

6. If f(x) = 3x⁷ - 5x⁴ + 2x - 8, what happens to f(x) as x approaches negative infinity?

Solution:

Degree: 7 (odd)

Leading coefficient: 3 (positive)

For odd degree with positive leading coefficient:

As x → -∞, f(x) → -∞

Key Takeaways

A polynomial function has the form f(x) = a_nxⁿ + ... + a₁x + a₀ with non-negative integer exponents
The degree is the highest power; the leading coefficient is its coefficient
End behavior is determined by the leading term: look at degree (even/odd) and leading coefficient (positive/negative)
Zeros are values where f(x) = 0; they correspond to x-intercepts
Multiplicity tells us if the graph crosses (odd) or touches/bounces (even) at a zero
To graph polynomials: find zeros, determine multiplicity, establish end behavior, and plot key points

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