Precalculus by Richard Wright

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2-03 Polynomial Equations

Mr. Wright teaches the lesson.

Summary: In this section, you will:

Identify polynomial functions.
Identify the end behavior.
Graph polynomial functions.
Write polynomial functions.

SDA NAD Content Standards (2018): PC.4.1, PC.5.3, PC.6.4

Pioneer Memorial Church. — Figure 1: credit (RW)

The membership of the Seventh-day Adventist church has been increasing over several years. Its worldwide membership is given in table 1.

Table 1 Source: https://www.adventist.org/statistics/
Year	2011	2012	2014	2015	2017	2018	2019
SDA Church Membership	17,479,890	17,881,491	18,479,257	19,126,438	20,727,347	21,414,779	21,556,837

The church's worldwide membership can be modeled by M(t) = -4862.5t⁴ + 85502.2t³ − 448881.3t² + 1204217t + 16648840 where t is the number of years since 2010. This model can be used to estimate membership counts for missing years and to estimate future or previous membership counts. Remember this models only the years within the data. The further the extrapolation is from the given data, the more error exists in the estimate.

Identify Polynomial Functions

The formula above is an example of a polynomial function which is a function that is a sum of terms in which the variable has non-negative, integer exponents.

Polynomial Functions

Let n be a non-negative integer. A polynomial function is a function that can be written in the form

f(x) = a_nxⁿ + ⋯ + a₂x² + a₁x + a₀

Each a_i is a coefficient and can be any real number.
a_n ≠ 0.
Each product a_ixⁱ is a term of a polynomial function.

Example 1: Identify Polynomial Functions

Which of the following are polynomial functions?

f(x) = x³ · 4x − 6x − 1
g(x) = 2x(5x + 4)
$h(x) = 4\sqrt{x} + 17$

Solution

The first two functions are examples of polynomial functions because they can be written in the form f(x) = a_nxⁿ + ⋯ + a₂x² + a₁x + a₀, where the powers are non-negative integers and the coefficients are real numbers.

f(x) can be written as f(x) = 4x⁴ − 6x − 1.
g(x) can be written as g(x) = 10x² + 8x.
h(x) cannot be written in this form and is therefore not a polynomial function.

Graphs of Polynomial Functions

A polynomial function can have an infinite variety in the number of terms and the exponent of the variable. Even though the the order of the terms of a polynomial function does not affect performing algebraic operations, the polynomial's terms are usually written in order of highest to lowest exponent. This is the general form. The degree of the polynomial is the highest exponent on the variable. If the polynomial is written in general form, the degree will be the first exponent of the variable. The leading coefficient is the coefficient of the term containing the highest exponent of the variable. If the polynomial is in general form, the leading coefficient is the first coefficient.

Terminology of Polynomial Functions

General form: The terms are written from highest to lowest exponent on the variable.
Degree: Highest exponent on the variable.
Leading coefficient: Coefficient of the term containing the highest exponent of the variable.

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.

Characteristics of Graphs of Polynomial Functions

The graph of a polynomial function is smooth and continuous. This means there are no sharp turns or breaks in the graph.

Graphs of Polynomials

The y-intercept is the point where the graph crosses the y-axis and can be found by substituting x = 0.
The x-intercepts are points where the graph crosses the x-axis and can be found by making the function equal to zero and solving for x.
- There are at most the same number of x-intercepts as the degree of the function.
- The x-intercepts reveal several other things about the function.
  1. If f(k) = 0, then k is a zero of f.
  2. If k is a zero, then it is a solution to f(x) = 0.
  3. If k is a zero, then (x – k) is a factor of f.
  4. If k is a real zero, then (k, 0) is a x-intercept.
- Multiplicity of zeros
  - If the graph crosses the x-axis, then the zero and factor occur an odd number of times.
  - If the graph touches the x-axis without crossing it, then the zero and factor occur an even number of times.
  - The number of times the zero and factor occur is called the zero's multiplicity. The total multiplicity should equal the degree of the function.
- The turning points are the places where the graph changes between increasing and decreasing. There are at most one less turning point than the degree of the function.
- End behavior describes what happens at the left and right ends of the graph of the functions.

Example 2: Analyze a Polynomial Graph

Given the graph in figure 2, find the

y-intercept
x-intercepts
real zeros of the function and describe their multiplicity
the number of turning points

Graph of f(x)=1/2 (x+2)(x-2)^2 — Figure 2

Solution

The y-intercept is where the graph crosses the y-axis which is (0, 4).
The x-intercepts occur where the graph intercepts the x-axis which are (−2, 0) and (2, 0).
The real zeros are the x-values of the x-intercepts. At the zero x = −2, the graph crosses over the x-axis, so it has an odd multiplicity, perhaps 1. At the zero x = 2, the graph does not cross over the x-axis, so it has an even multiplicity, perhaps 2. So the function could have a degree of 1 + 2 = 3.
The graph shows 2 turning points. Since there can be up to one less turning point than degree, the degree of this function could be 3.

Example 3: Find a Polynomial Function

Find the polynomial function whose x-intercepts are (−3, 0) multiplicity 2 and (2, 0) multiplicity 2 and y-intercept (0, 2).

Solution

Write each x-intercept as a factor in the form (x − k) where k are the x-intercepts. Use a as the leading coefficient. This will vertically stretch our function to pass through the y-intercept. The factors are squared because they have multiplicity 2.

f(x) = a(x − (−3))²(x − (2))²

Simplify.

f(x) = a(x + 3)²(x − 2)²

Substitute a point such as the y-intercept for x and f(x) and solve for a.

2 = a(0 + 3)²(0 − 2)²

2 = a(36)

$$ \frac{1}{18} = a $$

Substitute the value of a into the function and multiply it all out.

$$ f(x) = \frac{1}{18}(x + 3)^2 (x - 2)^2 $$

Square the factors.

$$ f(x) = \frac{1}{18}(x^2 + 6x + 9)(x^2 - 4x + 4) $$

Multiply the trinomials using the distributive property.

$$ f(x) = \frac{1}{18}(x^4 + 2x^3 - 11x^2 - 12x + 36) $$

Distribute the $\frac{1}{18}$.

$$ f(x) = \frac{1}{18}x^4 + \frac{1}{9}x^3 - \frac{11}{18}x^2 - \frac{2}{3}x + 2 $$

Try It 1

Given the graph of g(x) in Figure 4, find the

y-intercept
x-intercepts
the function.

Answer

(0, 3)
(−3, 0), (1, 0)
f(x) = −x² − 2x + 3

Example 4: Determine the Intercepts of a Polynomial Function

Given the polynomial function f(x) = (x + 2)(x + 3)(x − 4), 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 x = 0.

f(x) = (x + 2)(x + 3)(x − 4)

f(0) = (0 + 2)(0 + 3)(0 − 4)

f(0) = −24

The y-intercept is (0, −24).

The x-intercepts occur when the output is zero.

f(x) = (x + 2)(x + 3)(x − 4)

0 = (x + 2)(x + 3)(x − 4)

Because this is factored, use the Zero Product Theorem. Set each factor equal to zero and solve for all the x's.

x + 2 = 0 or x + 3 = 0 or x − 4 = 0

x = −2 or x = −3 or x = 4

The x-intercepts are (−2, 0), (−3, 0), and (4, 0).

These intercepts are on the graph of the function shown in figure 4.

Graph of f(x)=(x+2)(x+3)(x-4), which labels all the intercepts. — Figure 4

Example 5: Determine the Intercepts of a Polynomial Function with Factoring

Given the polynomial function f(x) = x⁴ − x² − 12, determine the y- and x-intercepts.

Solution

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

f(x) = x⁴ − x² − 12

f(0) = (0)⁴ − (0)² − 12

f(0) = −12

The y-intercept is (0, −12).

The x-intercepts occur when the output is zero.

f(x) = x⁴ − x² − 12

0 = x⁴ − x² − 12

Factor the trinomial.

0 = (x² − 4)(x² + 3)

Because this is factored, use the Zero Product Theorem. Set each factor equal to zero and solve for all the x's.

x² − 4 = 0 or x² + 3 = 0

x² = 4 or x² = −3

x = ±2 or $x = ±\sqrt{3}i$

x-intercepts are only real numbers, so the x-intercepts are (−2, 0) and (2, 0).

These intercepts are on the graph of the function shown in figure 5.

Graph of f(x)=x^4-x^2-12, which labels all the intercepts. — Figure 5

Try It 2

Given the polynomial function f(x) = 2x³ − 6x² − 20x, determine the y- and x-intercepts.

Answer

y-intercept (0, 0); x-intercepts (0, 0), (−2, 0)

Identify End Behavior of Polynomial Functions

Sometimes is it important to know what happens at the edges of a graph. The degree of a polynomial function and the leading coefficient are enough to provide patterns about the end behavior. End behavior is what the graph does on the left and right side of the graph. 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. See table 2.

Table 2
	+ Leading Coefficient	− Leading Coefficient
Even Degree
Odd Degree

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

Describe the end behavior and classify the polynomial function in figure 6.

Graph of an odd-degree polynomial. — Figure 6

Solution

The graph rises to the left and falls to the right. Comparing that to table 2. This graph has the shape of an odd degree polynomial function with a negative leading coefficient.

Try It 3

Describe the end behavior and classify the polynomial function in figure 7.

Graph of an even-degree polynomial. — Figure 7

Answer

The graph falls to both the left and right, so it is an even degree polynomial with a negative leading coefficient.

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

Given the function f(x) = 2x(x − 1)(x + 3), express the function as a polynomial in general form, and determine the leading coefficient, degree, and end behavior of the function.

Solution

Multiply the factors to get the general form for f(x).

f(x) = 2x(x − 1)(x + 3)

f(x) = 2x(x² + 2x − 3)

f(x) = 2x³ + 4x² − 6x

The leading coefficient is the coefficient of the variable with the highest exponent which is 2. The degree is the highest exponent which is 3. The degree is odd and the leading coefficient is positive, so by table 1, the graph falls to the left and rises to the right.

Try It 4

Given the function f(x) = −3(x + 1)(x + 2), express the function as a polynomial in general form, and determine the leading coefficient, degree, and end behavior of the function.

Answer

The general form is f(x) = −3x² − 9x − 6. The leading coefficient is −3. The degree is 2. The graph falls to the left and falls to the right.

Graph Polynomial Functions

To Graph a Polynomial Function

Find the x-intercepts.
Make a table of values around the x-intercepts.
Plot the points.
Draw a smooth curve through the points making sure the end behavior is correct.

Example 8: Graph a Polynomial Function

Graph f(x) = −x³ + 4x.

Solution

Find the x-intercepts by setting the function equal to zero and factoring.

0 = −x³ + 4x

0 = −x(x² − 4)

0 = −x(x − 2)(x + 2)

Make each factor = 0.

−x = 0 or x − 2 = 0 or x + 2 = 0

x = 0 or x = 2 or x = −2

The x-intercepts are (−2, 0), (0, 0), and (2, 0). Make a table of values around the x-intercepts.

x	−4	−3	−2	−1	0	1	2	3	4
y	48	15	0	−3	0	3	0	−15	−48

Plot the points and draw a smooth curve. The leading coefficient is negative (−1) and the degree is odd (3) so the end behavior should rise to the left and fall to the right.

Try It 5

Graph $f(x) = -\frac{1}{4}x^4 + \frac{5}{2}x^2 - \frac{9}{4}$.

Answer

Lesson Summary

Polynomial Functions

Let n be a non-negative integer. A polynomial function is a function that can be written in the form

f(x) = a_nxⁿ + ⋯ + a₂x² + a₁x + a₀

Each a_i is a coefficient and can be any real number.
a_n ≠ 0.
Each product a_ixⁱ is a term of a polynomial function.

Terminology of Polynomial Functions

General form: The terms are written from highest to lowest exponent on the variable.
Degree: Highest exponent on the variable.
Leading coefficient: Coefficient of the term containing the highest exponent of the variable.

Graphs of Polynomials

The y-intercept is the point where the graph crosses the y-axis and can be found by substituting x = 0.
The x-intercepts are points where the graph crosses the x-axis and can be found by making the function equal to zero and solving for x.
- There are at most the same number of x-intercepts as the degree of the function.
- The x-intercepts reveal several other things about the function.
  1. If f(k) = 0, then k is a zero of f.
  2. If k is a zero, then it is a solution to f(x) = 0.
  3. If k is a zero, then (x – k) is a factor of f.
  4. If k is a real zero, then (k, 0) is a x-intercept.
- Multiplicity of zeros
  - If the graph crosses the x-axis, then the zero and factor occur an odd number of times.
  - If the graph touches the x-axis without crossing it, then the zero and factor occur an even number of times.
  - The number of times the zero and factor occur is called the zero's multiplicity. The total multiplicity should equal the degree of the function.
- The turning points are the places where the graph changes between increasing and decreasing. There are at most one less turning point than the degree of the function.
- End behavior describes what happens at the left and right ends of the graph of the functions.

To Graph a Polynomial Function

Find the x-intercepts.
Make a table of values around the x-intercepts.
Plot the points.
Draw a smooth curve through the points making sure the end behavior is correct.

Helpful videos about this lesson.

Practice Exercises

What is the end behavior of a polynomial function with odd degree if the leading coefficient is positive?
If the graph of a polynomial just touches the x-axis and then changes direction, what can be concluded about the factored form of the polynomial?

Find the degree and leading coefficient for the given polynomial.

g(x) = −2x² + 4x⁴ − 5x

Describe the end behavior of the functions.

f(x) = −x⁴ + x³
g(x) = 2x³ + 3x² + 5x − 17

Find the intercepts of the functions.

h(t) = 3(t + 2)(t − 1)(t + 3)
f(x) = x⁴ − 81
g(r) = r³ + 3r² − 10r

Determine the least possible degree of the polynomial function shown.

Graph the polynomial function using a graphing calculator. Based on the graph, determine the intercepts and the end behavior.

f(x) = x²(x + 2)(x − 2)
f(x) = −2x⁴ − 4x³

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The y-intercept is (0, 4). The x-intercepts are (−2, 0), (2, 0). Degree is 2. End behavior: Falls to the left and falls to the right.
The y-intercept is (0, 0). The x-intercepts are (0, 0), (3, 0). Degree is 3. End behavior: Rises to the left, falls to the right.

Find the zeros and give the multiplicity of each.

f(x) = x⁴ + 6x³ + 5x²
f(x) = −2x⁴ + 20x³ − 50x²

Graph the polynomial functions. Identify the x- and y-intercepts, multiplicity, and end behavior.

g(x) = (x + 3)(x − 1)²
k(x) = (x − 3)(x + 2)²

Use the graph to write the polynomial function of least degree.

Use the given information about the polynomial graph to write the function.

Degree 5. Zeros of multiplicity 2 at x = 2 and x = −1, and a zero of multiplicity 1 at x = 3. y-intercept at (0, 6).

Problem Solving: Use the written statements to construct a polynomial function that represents the required information.

An ripple is expanding as a circle on a pond when a pebble was thrown into it. The radius of the circle is increasing at the rate of 8 inches per second. Express the area of the circle as a function of t, the number of seconds after the pebble hit the water.
A rectangle has a length of 20 units and a width of 16 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x.

Mixed Review

(2-02) Sketch the graph of f(x) = −x² − 3x + 4.
(2-02) Rewrite the following quadratic function in standard form and give the vertex. g(x) = 2x² − 28x + 90
(2-01) Solve 0 = x² − 4x + 5.

Answers

Falls to the left, rises to the right.
There will be a factor raised to an even power.
Degree: 4, Coefficient: 4
Falls to the left, falls to the right
Falls to the left, rises to the right
y-intercept is (0, −18); t-intercepts are (−3, 0), (−2, 0), and (1, 0)
y-intercept is (0, −81); x-intercepts are (3, 0) and (−3, 0)
y-intercept is (0, 0); x-intercepts are (−5, 0), (0, 0), and (2, 0)
5
4
; y-intercept (0, 0); x-intercepts (−2, 0), (0, 0), (2, 0); Rises to the left, rises to the right
; y-intercept (0, 0); x-intercepts (−2, 0), (0, 0); Falls to the left, falls to the right
f(x) = −x² + 4
f(x) = −x³ + 3x²
−5 with multiplicity 1, −1 with multiplicity 1, 0 with multiplicity 2
5 with multiplicity 2, 0 with multiplicity 2
; x-intercepts (−3, 0) with multiplicity 1 and (1, 0) with multiplicity 2; y-intercept (0, 3); Falls to the left, rises to the right.
; x-intercepts (3, 0) with multiplicity 1, (−2, 0) with multiplicity 2; y-intercept (0, −12); Falls to the left, rises to the right.
f(x) = −x⁴ − x³ + 2x²
$f(x) = -\frac{1}{2}x^5 + \frac{5}{2}x^4 - \frac{3}{2}x^3 - \frac{13}{2}x^2 + 4x + 6$
A = 64πt²
f(x) = 4x³ − 72x² + 320x
g(x) = 2(x − 7)² − 8
x = 2 − i, 2 + i