Learning activity 1.2: Describing polynomial expressions

As you noticed in the introduction, there are many different types of functions that can be used to model different types of real life situations. For example, you have to think about,” What function can be used to model the height of the football above the ground over time?”

Understanding what a polynomial function is and what the equation of a polynomial function looks like can help you understand the structures around you. There are a lot of different types of designers in the world, and they use polynomial functions when designing things like bridges, roller coasters, and packaging containers, to name just a few. To understand how designers use their knowledge of polynomial functions to their advantage, we need to understand their language. For the next 9 activities, we will be learning the language of polynomial functions and then use our knowledge to sketch polynomial functions that are used by designers to design things like roller coasters.

In this Lesson, we are going to start by learning about Polynomial Expressions. We will also understand the difference between Polynomial Expression and Functions. It is important to recognize one variable polynomial functions and also how to determine the degree of any given polynomial function since the degree of a polynomial function affects the behaviour and appearance of the polynomial function.

Let us just start by first understanding the difference between a Polynomial Expression and a Function.

Definition: A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Definition: An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide).

Example: 7a+2b+3c is clearly an algebraic expression but it does not have to be a function, since the letters a,b,c don't have to be inputs. But if a is an input, then we would have a function: f(a)=7a+2b+3cf, but b and c would be constants.

Example: f(x)=4x+9 with x∈N is a function because it has a set of inputs (x) and a set of outputs (numbers of the form 4x+9).

But 4x+9 is also an expression as it contains numbers, variables and operators. Keep in mind that, all functions (at least the formulas represented by them) are expressions, but not all expressions are functions.

A one variable polynomial expression is an expression that is made up of only one variable (that is, one letter of the alphabet)

The polynomial expression: $7 x^{2} + x - 3$

is a one variable polynomial expression because the only variable in the expression is $x$

A term is a number, variable, or the product of a number and variable.

The polynomial expression is made up of $3$ terms. The 3 terms are $7 x^{2}$ , $x$ and $- 3$

The number that is multiplied by the variable is called a coefficient

A term that contains only a number is called a constant

The degree of a polynomial expression is the highest exponent of the expression

If there is no visible exponent or coefficient on a term, it means that the exponent or coefficient is one.

If there is no visible variable on a term, it means that the variable has a zero exponent.

A polynomial expression in factored form is an expression written as a product of its factors.

For example, the factored form of the expression $x^{2} + x - 6$ is $(x + 3) (x - 2)$

Guided exploration

The purpose of this investigation is to compare the features of polynomial expressions to features of non-polynomial expressions

Look carefully at the following table:

Polynomial expressions	Expressions that are not polynomials
$0.4 x - 3$	$\sqrt{x} - 1$
$\frac{2}{7} x - 1$	$\frac{7 x}{- 2 x^{2} + 5 x - 1}$
$- x^{2} + 5 x - 1$	$\sin (x - 1)$
$8 (x - 1) (x + 19)$	$3^{x} - 1$
$- 3 {(x - 12)}^{2}$	${- x}^{- 2} + 5 x - 1$
$4 x^{3} + 2 x^{2} - x + 47$	$4 x^{- 3} + 2 x^{2} - x + 47$
$- x^{4} + 6 x^{3} + 10 x - 17$	${(x - 1)}^{0.5}$
${(x - 23)}^{2} (x + 11)$	$\sqrt{{(x - 23)}^{2} (x + 11)}$
${(x - 1)}^{4}$	$e^{x - 3}$
${(x - 3)}^{2} {(x + 5)}^{2}$	$\log (2 x + 1)$

What do you notice about the expressions in the two columns?
What’s the same?
What’s different?

Polynomial expressions	Expressions that are not polynomials
Exponents are positive (including zero) Exponents are whole numbers The bases are variables The variable is uniform (the same) Some expressions are in standard form and some are in factored form	Some of the exponents are negative Some of the exponents are fractions Some of the terms are divided by a variable Some of the expressions have words like sine and log Some expressions have a number base and a variable exponent

Polynomial expressions

Expressions that are not polynomials

Exponents are positive (including zero)

Exponents are whole numbers

The bases are variables

The variable is uniform (the same)

Some expressions are in standard form and some are in factored form

Some of the exponents are negative

Some of the exponents are fractions

Some of the terms are divided by a variable

Some of the expressions have words like sine and log

Some expressions have a number base and a variable exponent

Notebook

Use your observations to make up a personalized definition of a polynomial expression. It doesn’t have to be a sentence; it could be in point form, pictures, phrases or any other expression that makes sense for you. Record it in your notebook.

Summary

A one variable polynomial expression can be in standard form or factored form

A one variable polynomial expression is a series (sum) of terms where each term is the product of a constant and a variable with an exponent that is a positive whole number when written in standard form

Examples:

1. Determine whether each of the following expressions is a polynomial expression or not. Explain your thought process.

$a) \sqrt{2 x - 5}$

$\sqrt{2 x - 5}$ is not a polynomial because $\sqrt{2 x - 5} = {(2 x - 5)}^{\frac{1}{2}}$ , therefore the exponent is a fraction $(\frac{1}{2})$ , not a whole number

$b) {(7 x + 9)}^{2} (1 - x)$

${(7 x + 9)}^{2} (1 - x)$ is a polynomial because all the terms inside the brackets have exponents that are positive whole numbers and the brackets also have exponents that are positive whole numbers. If we expand ${(7 x + 9)}^{2} (1 - x)$ we get a series of terms with exponents that are positive whole numbers

$c) x - {2 x}^{- 3}$

$x - {2 x}^{- 3}$ is not a polynomial because the term ${2 x}^{- 3}$ has a negative exponent (-3)

$d) \frac{x}{x + 7}$

“ $\frac{x}{x + 7}$ is not a polynomial because it is a rational expression” or

“...is not a polynomial because it involves dividing by a variable” or

“...is not a polynomial because there is a variable in the denominator” or

“...is not a polynomial because it could be written with a negative exponent

$x {(x + 7)}^{- 1}$ ”

2. State the degree of each polynomial expression. Explain your thought process.

$a) x + 9$

degree = 1 since the highest exponent on the polynomial expression is 1

Think: $x^{1} + 9$

No exponent means the exponent is $1$

$b) 16 x^{2} - x + 9$

degree = 2 since the highest exponent on the polynomial expression is 2

$c) 5 x^{3} + 8 x^{2} - 3 x + 9$

degree = 3 since the highest exponent on the polynomial expression is 3

${d) 2 x}^{4} - 11 x^{2} + x - 23$

degree = 4 since the highest exponent on the polynomial expression is 4

Practice Questions

1. Determine whether each of the following expressions is a polynomial expression or not. Explain your thought process.

a) $\frac{1}{x^{2} - 10 x}$

“is not a polynomial because it is rational expression” or

“...is not a polynomial because it involves dividing by a variable” or

“...is not a polynomial because there is a variable in the denominator”

“...a fraction (cannot be expressed as a sum of a series of terms)”

b) $9 x^{- 1} + 3 x$

is not a polynomial because the term ${9 x}^{- 1}$ has a negative exponent (-1)

c) ${(3 x - 7)}^{2} (x + 33)$

is a polynomial because all the terms inside the brackets have exponents that are positive whole numbers and the brackets also have exponents that are positive whole numbers. If we expand ${(3 x - 7)}^{2} (x + 33)$ we get a series of terms with exponents that are positive whole numbers

d) $\sqrt{7 x + 13}$

is not a polynomial because $\sqrt{7 x + 13} = {(7 x + 13)}^{\frac{1}{2}}$ , therefore the exponent is a fraction, not a whole number

e) ${- 11 x}^{3} - 9 x^{2} + 7 x - 21$

a polynomial because all the terms have exponents that are positive whole numbers

2. State the degree of each polynomial expression. Explain your thought process.

a) $2 x^{4} + 9 x^{2} - 11$

degree = $4$ since the highest exponent on the polynomial expression is $4$

b) $6 x - 9$

Think: ${6 x}^{1} - 9$

No exponent means the exponent is $1$

degree = 1 since the highest exponent on the polynomial expression is 1

c) $x^{3} + 8 x^{2} - 3 x + 9$

degree = $3$ since the highest exponent on the polynomial expression is $3$

d) $13 x^{2} - 9 x + 100$

degree = $2$ since the highest exponent on the polynomial expression is 2

Additional practice

Please click on the following link for more practice and self assessment:

Introduction to Polynomials (Opens in new window)

Join the discussion

What strategies did you use to answer the questions in this self-assessment activity? Explain how you knew what to do, and any challenges you had when solving.

Reflection Question

Notebook

Answer the following question and check your answer when ready.

Rod says that he is thinking of a one variable polynomial expression that has the following characteristics:

has degree of 4
has 3 terms
has one of the terms with a degree of 1
one of the terms has a coefficient of -5

What could Rod’s polynomial expression be? Explain your thought process.

(Note: there is more than one right answer)

Your reflection in your notebook could be contained inside something like these blank tables.

Thought process	Steps

Join the discussion

You are becoming more familiar with the characteristics of a polynomial expression.

Create your own question similar to Rod’s. Select a one variable polynomial expression, and share some of its characteristics. Reply to at least two of your peers’ posts with a polynomial that fits their characteristics.

When someone replies to your question, let them know if they matched yours exactly.

Sample Answer

Step 1: I choose the variable $x$ .

Thought process: The polynomial is a one variable polynomial expression, therefore I need to choose a variable to use

Step 2: The first term can be $x^{4}$ .

Thought process: Since the polynomial expression has a degree of 4, it means that the highest exponent on Rod’s polynomial expression is 4.

Step 3: The second term can be $x$ .

Thought process: One of the terms has degree 1, which means the exponent of the term is 1.

Step 4: I choose my third term to have a coefficient of $- 5$ and a degree of $2$ : $- 5 x^{2}$ .

Thought process: One term has a coefficient of $- 5$ , which means $x^{4}$ or $x$ or a third term of my choice can have a coefficient of $- 5$ since my expression is supposed to have 3 terms.

Step 5: My expression is $x^{4} - 5 x^{2} + x$ .

Thought process: When I combine all my terms using addition or subtraction, I write the terms in descending order.

In the next learning activity, we will learn to identify polynomial functions using graphs and tables of values