Learning activity 1.3: Polynomial functions in factored form

Factors are numbers that are multiplied together to give a product. For example, 3 and 4 are factors of 12. What are some other factors of 12?

1 and 12
2 and 6
2 and 2 and 3

You can also find the factors of polynomial expressions. For example, $(x + 3) a n d (x + 1)$ are the factors of $x^{2} + 4 x + 3$ .

Complete the following questions, recording your answers in your notebook. Then, check your work with the suggested answers included further below.

Question 1

What polynomial could $(x - 4) a n d (x + 2)$ be factors of?

Question 2

What polynomial could $(x - 1) a n d (x + 5)$ be factors of?

Suggested Answers

$(x - 4) (x + 2)$

${= x}^{2} - 4 x + 2 x - 8$

$= x^{2} - 2 x - 8$

$(x - 4) a n d (x + 2)$ are factors of $x^{2} - 2 x - 8$

$(x - 1) (x + 5)$

${= x}^{2} - x + 5 x - 5$

$= x^{2} + 4 x - 5$

$(x - 1) a n d (x + 5)$ are factors of $x^{2} + 4 x - 5$

Topic 1 – Factored form and the $x$ -intercepts of a polynomial function

In the previous learning activity, you learned about polynomial functions in standard form:

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + ... + a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$

What information does a polynomial function in standard form tell you? Refer to the suggested answers to check your ideas.

Leading coefficient
Leading term
Degree of the function

Another way to express polynomial functions is in factored form.

Factored form of a polynomial function

$f (x) = a (x - c_{1}) (x - c_{2}) \dots (x - c_{n})$

$a :$ leading coefficient

$c_{1}, c_{2}, \dots, c_{n} :$ constant, can be any real number

INVESTIGATION: Relating factored form and $x$ -intercepts on a graph

Transferable skills

Use your notebook (or use your internet search engine to find an online graphing tool) to graph the following polynomial functions and answer the questions about the key features. When you have finished, check your ideas with a classmate.

1. $y = 0.2 (x + 3) (x - 1) (x - 2)$

Graph the function

Identify each of the following:

Sign of the leading coefficient
Degree of polynomial
$x$ -intercepts

2. $y = - (x + 2) (x - 2) (x - 4)$

Graph the function

Identify each of the following:

Sign of the leading coefficient
Degree of polynomial
$x$ -intercepts

3. $y = (x + 3) (x) (x - 1) (x - 2)$

Graph the function

Identify each of the following:

Sign of the leading coefficient
Degree of polynomial
$x$ -intercepts

Summary

Let’s summarize how the equation of a polynomial function in factored form relates to the graph. Check your ideas with the suggested answers.

How do the $x$ -intercepts of the graph relate to the equation of the polynomial function?

They are the constant term, $c_{n}$ , in the formula. Notice that they are the number being subtracted from $x$ in the factors.

For example, for a polynomial equation with factor ( $x - 2$ ), an $x$ -intercept would be (positive) $2$ . For a polynomial equation with factor ( $x + 3$ ), an $x$ -intercept would be - $3$ (negative 3). Many students find it helpful to determine the $x$ -intercept by writing down what appears after the $x$ , but with the opposite sign.

How could you solve for an $x$ -intercept when given factored form algebraically?

You could set the factor equal to zero and solve for $x$

For example, for the polynomial equation with factor ( $x - 2$ ), we could solve
$x - 2 = 0$
$x = 0 + 2$
$x = 2$

Sketching polynomial functions

Now that you understand how the $x$ -intercepts relate to the equation of the function, you can also graph the functions using those $x$ -intercepts and the end behaviours.

Examples:

Let’s go through an example of graphing a polynomial function given the key features. Then, you’ll have a chance to practice.

A graph has a degree $4$ , a negative leading coefficient and $x$ -intercepts at $- 2, - 1, 2, 3$

Since this is an even degree function with a negative leading coefficient, the graph will start and end low.

Therefore, the end behaviours of this graph are:

Start low

End low

How would you sketch the function based on the end behaviours and the $x$ -intercepts? Select each tab below to learn more.

Plot the $x$ -intercepts on a graph

Draw in the end behaviours

Since it is an even degree function (i.e. degree of 4) with a negative leading coefficient, the graph will start and end low.

Connect the other $x$ -intercepts by alternative curves of $\cap$ and $\cup$ shapes

The graph may be a bit different than this one. The peaks may be lower or higher and the valleys may be lower or higher. As long as the general shape is the same, it is a reasonable graph for the given key features.

Now, assume that a graph has a degree $3$ , a positive leading coefficient and $x$ -intercepts at $- 3, 1, 4$ . What will the end behaviours of this graph be? Provide your answer below and then check the suggested answer.

Start low

End high

Since this is an odd degree function with a positive leading coefficient, the graph will start low and end high.

In your notebook sketch the function based on the end behaviours and the $x$ -intercepts by completing each of the following steps:

Step 1: Plot the $x$ -intercepts on a graph

Step 2: Draw in the end behaviours

Step 3: Connect the other $x$ -intercepts by alternative curves of $\cap$ and $\cup$ shapes

You can check your ideas by reviewing the suggested answers included in the following tabs:

Suggested Answers

Once you have sketched the function based on the end behaviours and the $x$ -intercepts, review each tab to check your answers.

Step 1: Plotting the $x$ -intercepts would produce the following graph:

Step 2: Drawing in the end behaviours would produce the following graph:

Since it is an odd degree function (i.e. degree $3$ ) with a positive leading coefficient, the graph will start low and end high.

Step 3: Connecting the other $x$ -intercepts by alternative curves of $\cap$ and $\cup$ shapes could produce the following graph:

Your graph may be a bit different than this one. The peaks may be lower or higher and the valleys may be lower or higher. As long as the general shape is the same, it is a reasonable graph for the given key features.

Notebook

Great work! Now, complete the “Practice Activity: Graphing a polynomial function given the key features (Opens in new window)” and then check your solutions using the “Suggested Answers: Graphing a polynomial function given the key features (Opens in new window)”.

Factored form of polynomial functions

Select the correct answer for each of the following practice questions on the factored form of polynomial functions.

Topic 2 – Roots and zeros of polynomial functions

The zeros of a polynomial function correspond to the $x$ -intercepts of the graph. They are also equal to the roots of the equation when $f (x) = 0$ .

Study the graph, considering why it makes sense that the zeros, or $x$ -intercepts, correspond to the roots for $f (x) = 0 .$

The $x$ -intercepts on a graph correspond to the point where the graph touches the
$x$ -axis, or where the $y$ -value of a point is 0. Setting the equation $f (x)$ equal to zero, is the same as substituting $0$ in for $y$ and solving for $x$ .

The terms zero and root are sometimes used interchangeably. However, zeros are associated with a function and roots are associated with the solution to an equation. Roots and zeros are the same when an equation is formed by substituting 0 for f(x) in the general equation of a function.

Review:

Previously, you learned how to identify the $x$ -intercepts of a polynomial function given the formula. Let’s go through an example; select each tab below to learn more.

First, we determine the $x$ -intercepts of the function $f (x) = (x - 3) (x + 2) (x + 4),$

by setting the factors equal to zero and then solving for $x$ . For example:

x - 3 = 0

x + 2 = 0

x + 4 = 0

x = 0 + 3

x = 0 - 2

x = 0 - 4

x = 3

x = - 2

x = - 4

The $x$ -intercepts are $3, - 2, a n d - 4$

Next, we substitute in the $x$ -intercepts, one at a time to determine what the equation will equal. For example:

$f (x) = (x - 3) (x + 2) (x + 4)$

$f (3) = (3 - 3) (3 + 2) (3 + 4)$

$f (3) = (0) (5) (7)$

$f (3) = 0$

$f (x) = (x - 3) (x + 2) (x + 4)$

$f (- 2) = (- 2 - 3) (- 2 + 2) (- 2 + 4)$

$f (- 2) = (- 5) (0) (2)$

$f (- 2) = 0$

$f (x) = (x - 3) (x + 2) (x + 4)$

$f (- 4) = (- 4 - 3) (- 4 + 2) (- 4 + 4)$

$f (- 4) = (- 7) (- 2) (0)$

$f (- 4) = 0$

Did you notice? When substituting any of the $x$ -intercepts, the equation, $f (x)$ , equals zero, hence the name ‘zero’ for the $x$ -intercept.

Zeros of order $n$

For a polynomial with a factor of $(x - a)$ , $x = a$ is a zero or root of the equation.

If a polynomial has a factor that is repeated $n$ times (has an exponent $n$ ), the zero is of order $n$ .

For example, $f (x) = (x - 2) (x + 1)^{2}$ has a zero of - $1$ with an order of $2$ , and a zero of $2$ .

Examples:

For each of the following functions, determine all of the zeros, along with their orders. Select each function to check the suggested answer.

This has a zero of - $3$ , and a zero of $2$ with an order of $2$ .

This has a zero of $1$ , and a zero of $5$ with an order of $3$ .

This has a zero of - $4$ with an order of $2$ , and a zero of $1$ with an order of $3$ .

Transferable skills

INVESTIGATION: Zeros of order $n$ on a graph

Complete the Zeros of order $n$ on a graph practice exercise. Use your notebook, Geogebra, or the online graphing tool of your choice. When finished, use the “Zeros of Order $n$ on a Graph Suggested Answers” to check your work.

Zeros of order $n$ on a graph practice exercise

Graph each of the following polynomial functions, answering the questions about the key features.

Functions:

$1 . f (x) = (x + 2) (x - 1)^{2}$

2. $f (x) = (x + 1) (x + 3)^{3}$

$3 . f (x) = (x - 5) (x - 2) (x + 3)^{2}$

$4 . f (x) = (x - 2)^{2} (x + 1)^{2}$

Questions:

a. Graph the function.

b. Identify the zeros and their order.

c. What happens to the graph at each zero?

Zeros of Order n on a Graph Suggested Answers (Opens in new window)

Sketching polynomial functions with zeros of order $n$

Now that you understand how the $x$ -intercepts and their orders relate to the graph of the function, you can also sketch the functions using those $x$ -intercepts and the end behaviours.

Examples:

Let’s go through an example of graphing a polynomial function with zeros of order $n$ . Check your solutions with the suggested answers.

$y = {(x - 1)}^{2} (x + 2)$

We must determine how many factors include an $x$ . The first factor has an exponent of 2, so two factors include an $x$ . Additionally, the third factor contains an x.
$(x) (x) (x) = x^{3}$

This polynomial has a degree of 3.

Start low

End high

Since this is an odd degree function with a positive leading coefficient of 1, the graph will start low and end high.

Zero (of order 2) at $x = 1$ and a zero (of order 1) at $x = - 2$

x = 1

, the zero of order 2, the graph will touch the

x

-axis, but will not cross it. At

x = - 2

, the graph will cross the

x

-axis

Now, let’s sketch the function $y = {(x - 1)}^{2} (x + 2)$ based on the end behaviour and the $x$ -intercepts and their orders.

Since it is an odd degree function (i.e. degree of 3) with a positive leading coefficient, the graph will start low and end high

Connect the $x$ -intercepts, paying attention to whether the graph will cross or touch the x-axis based on the order of the zeros

Notebook

Here are two additional examples to try on your own.

Study each of the following functions, answering the questions in your notebook.

$1 . y = 0.5 (x + 2) (x + 1) (x - 1)^{2}$

2. $y = - 0.5 (x + 2) (x - 3)^{3}$

a) What is the degree of this function? You will need to count every factor that contains an $x$ .

b) What will the end behaviours of this graph be?

c) Identify the zeros and their orders.

d) How will the graph appear at the zeros?

Then, sketch each function based on the end behaviours and the $x$ -intercepts and their orders, following these steps:

Step 1: Plot the $x$ -intercepts on a graph

Step 2: Draw in the end behaviours

Step 3: Connect the $x$ -intercepts, paying attention to whether the graph will cross or touch the $x$ -axis, based on the order of the zeros.

Try It!

Complete these practice questions on the zeros of polynomial functions. Select the equation that corresponds with each graph.

Question 1

Question 2

Question 1: Suggested Answer

Question 2: Suggested Answer

Topic 1 – Factored form and the x -intercepts of a polynomial function

Transferable skills

Summary

Sketching polynomial functions

Examples:

Notebook

Topic 2 – Roots and zeros of polynomial functions

Examples:

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

2. f x = ( x - 1 ) ( x - 5 ) 3

3. f x = ( x + 4 ) 2 ( x - 1 ) 3

Transferable skills

Examples:

1. What is the degree of this function?

2. What will the end behaviours of this graph be?

3. Identify the zeros and their orders

4. What will happen to the graph at the zeros?

Step 1: Plot the x -intercepts on a graph

Step 2: Draw in the end behaviours

Step 3: Connect the x -intercepts

Notebook

Try It!

Topic 1 – Factored form and the $x$ -intercepts of a polynomial function