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### Course: Algebra (all content) > Unit 10

Lesson 36: Symmetry of polynomial functions# Symmetry of polynomials

Learn how to determine if a polynomial function is even, odd, or neither.

#### What you should be familiar with before taking this lesson

A function is an $y$ -axis.

**even function**if its graph is symmetric with respect to theAlgebraically, $f$ is an even function if $f(-x)=f(x)$ for all $x$ .

A function is an

**odd function**if its graph is symmetric with respect to the origin.Algebraically, $f$ is an odd function if $f(-x)=-f(x)$ for all $x$ .

If this is new to you, we recommend that you check out our intro to symmetry of functions.

#### What you will learn in this lesson

You will learn how to determine whether a polynomial is even, odd, or neither, based on the polynomial's equation.

## Investigation: Symmetry of monomials

A $f(x)=a{x}^{n}$ where $a$ is a real number and $n$ is an integer greater than or equal to $0$ .

**monomial**is a one-termed polynomial. Monomials have the formIn this investigation, we will analyze the symmetry of several monomials to see if we can come up with general conditions for a monomial to be even or odd.

In general, to determine whether a function $f$ is even, odd, or neither even nor odd, we analyze the expression for $f(-x)$ :

- If
$f(-x)$ *is the same as* , then we know$f(x)$ is even.$f$ - If
$f(-x)$ *is the opposite of* , then we know$f(x)$ is odd.$f$ - Otherwise, it is neither even nor odd.

As a first example, let's determine whether $f(x)=4{x}^{3}$ is even, odd, or neither.

Here $f(-x)=-f(x)$ , and so function $f$ is an odd function.

Now try some examples on your own to see if you can find a pattern.

### Concluding the investigation

From the above problems, we see that if $f$ is a monomial function of $f$ is an $f$ is a monomial function of $f$ is an

*even*degree, then function*even function*. Similarly, if*odd*degree, then function*odd function.*Even Function | Odd Function | |
---|---|---|

Examples | ||

In general |

This is because $(-x{)}^{n}={x}^{n}$ when $n$ is even and $(-x{)}^{n}=-{x}^{n}$ when $n$ is odd.

This is probably the reason why even and odd functions were named as such in the first place!

## Investigation: Symmetry of polynomials

In this investigation, we will examine the symmetry of polynomials with more than one term.

### Example 1: $f(x)=2{x}^{4}-3{x}^{2}-5$

To determine whether $f$ is even, odd, or neither, we find $f(-x)$ .

Since $f(-x)=f(x)$ , function $f$ is an even function.

Note that all the terms of $f$ are of an even degree.

### Example 2: $g(x)=5{x}^{7}-3{x}^{3}+x$

Again, we start by finding $g(-x)$ .

At this point, notice that each term in $g(-x)$ is the $g(x)$ . In other words, $g(-x)=-g(x)$ , and so $g$ is an odd function.

*opposite*of each term inNote that all the terms of $g$ are of an odd degree.

### Example 3: $h(x)=2{x}^{4}-7{x}^{3}$

Let's find $h(-x)$ .

*opposite*of

Mathematically, $h(-x)\ne h(x)$ and $h(-x)\ne -h(x)$ , and so $h$ is neither even nor odd.

Note that $h$ has one even-degree term and one odd-degree term.

### Concluding the investigation

In general, we can determine whether a polynomial is even, odd, or neither by examining each individual term.

General rule | Example polynomial | |
---|---|---|

Even | A polynomial is even if each term is an even function. | |

Odd | A polynomial is odd if each term is an odd function. | |

Neither | A polynomial is neither even nor odd if it is made up of both even and odd functions. |

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