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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 even function if its graph is symmetric with respect to the

y

-axis.

Algebraically,

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 monomial is a one-termed polynomial. Monomials have the form

f (x) = a x^{n}

where

a

is a real number and

n

is an integer greater than or equal to

0

In 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 $f (x)$ ‍, then we know $f$ ‍ is even.
If $f (- x)$ ‍ is the opposite of $f (x)$ ‍, then we know $f$ ‍ is odd.
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.

$\begin{aligned} f (- x) & = 4 (- x)^{3} \\ = 4 (- x^{3}) & (- x)^{3} = - x^{3} \\ = - 4 x^{3} & Simplify \\ = - f (x) & Since f (x) = 4 x^{3} \end{aligned}$ ‍

Here

f (- x) = - f (x)

, and so function

f

is an odd function.

The graph of

y = f (x)

is symmetric with respect to the origin, which confirms our solution!

An cubic function on an x y coordinate plane. Its middle point is at (zero, zero). As x goes to negative infinity, y goes to negative infinity. As x goes to infinity, y goes to infinity. The graph is concave down from the interval negative infinity to zero. The graph is concave up from the interval zero to infinity.

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

1) Is $g (x) = 3 x^{2}$ ‍ even, odd, or neither?

2) Is $h (x) = - 2 x^{5}$ ‍ even, odd, or neither?

Concluding the investigation

From the above problems, we see that if

f

is a monomial function of even degree, then function

f

is an even function. Similarly, if

f

is a monomial function of odd degree, then function

f

is an odd function.

	Even Function	Odd Function
Examples	$g (x) = 3 x^{2}$ ‍	$h (x) = - 2 x^{5}$ ‍
In general	$f (x) = a x^{n}$ ‍ where $n$ ‍ is $even$ ‍	$f (x) = a x^{n}$ ‍ where $n$ ‍ is $odd$ ‍

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)

\begin{aligned} f (- x) & = 2 (- x)^{4} - 3 (- x)^{2} - 5 \\ = 2 (x^{4}) - 3 (x^{2}) - 5 & (- x)^{n} = x^{n} when n is even \\ = 2 x^{4} - 3 x^{2} - 5 & Simplify \\ = f (x) & Since f (x) = 2 x^{4} - 3 x^{2} - 5 \end{aligned}

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)

\begin{aligned} g (- x) & = 5 (- x)^{7} - 3 (- x)^{3} + (- x) \\ = 5 (- x^{7}) - 3 (- x^{3}) + (- x) & (- x)^{n} = - x^{n} when n is odd \\ = - 5 x^{7} + 3 x^{3} - x & Simplify \end{aligned}

At this point, notice that each term in

g (- x)

is the opposite of each term in

g (x)

. In other words,

g (- x) = - g (x)

, and so

g

is an odd function.

Note 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)

\begin{aligned} h (- x) & = 2 (- x)^{4} - 7 (- x)^{3} \\ = 2 (x^{4}) - 7 (- x^{3}) & (- x)^{4} = x^{4} and (- x)^{3} = - x^{3} \\ = 2 x^{4} + 7 x^{3} & Simplify \end{aligned}

2 x^{4} + 7 x^{3}

is not the same as

h (x)

nor is it the opposite of

h (x)

Mathematically,

h (- x) \neq h (x)

and

h (- x) \neq - 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.	$f (x) = 2 x^{4} - 3 x^{2} - 5$ ‍
Odd	A polynomial is odd if each term is an odd function.	$g (x) = 5 x^{7} - 3 x^{3} + x$ ‍
Neither	A polynomial is neither even nor odd if it is made up of both even and odd functions.	$h (x) = 2 x^{4} - 7 x^{3}$ ‍

Check your understanding

3) Is $f (x) = - 3 x^{4} - 7 x^{2} + 5$ ‍ even, odd, or neither?

To start, let's decide if each term of

f (x) = - 3 x^{4} - 7 x^{2} + 5

is an even or odd function.

$- 3 x^{4}$ ‍ is an even function, since it's a monomial of $even$ ‍ degree.
$- 7 x^{2}$ ‍ is an even function, since it's a monomial of $even$ ‍ degree.
$5$ ‍ is an even function since it is a monomial of $even$ ‍ degree. Notice it can be written as $5 \cdot x^{0}$ ‍.

Since each term in the polynomial function is itself an even function (even degree), the polynomial is also even.

We could also see this algebraically by showing that

f (- x) = f (x) .

\begin{aligned} f (- x) \\ = - 3 (- x)^{4} - 7 (- x)^{2} + 5 \\ = - 3 (x^{4}) - 7 (x^{2}) + 5 & (- x)^{n} = x^{n} for even n \\ = - 3 x^{4} - 7 x^{2} + 5 & Simplify \\ = f (x) \end{aligned}

4) Is $g (x) = 8 x^{7} - 6 x^{3} + x^{2}$ ‍ even, odd, or neither?

To start, let's decide if each term of

g (x) = 8 x^{7} - 6 x^{3} + x^{2}

is an even or odd function.

$8 x^{7}$ ‍ is an odd function, since it's a monomial of $odd$ ‍ degree.
$- 6 x^{3}$ ‍ is an odd function, since it's a monomial of $odd$ ‍ degree.
$x^{2}$ ‍ is an even function, since it is a monomial of $even$ ‍ degree.

Since the terms in the polynomial are mixed (even and odd), then the polynomial is neither even nor odd.

We can also verify this algebraically. Notice that

g (- x)

is neither

g (x)

nor its opposite.

\begin{aligned} g (- x) & = 8 (- x)^{7} - 6 (- x)^{3} + (- x)^{2} \\ = 8 (- x^{7}) - 6 (- x^{3}) + x^{2} \\ = - 8 x^{7} + 6 x^{3} + x^{2} & Simplify \end{aligned}

5) Is $h (x) = 10 x^{5} + 2 x^{3} - x$ ‍ even, odd, or neither?

To start, let's decide if each term of

h (x) = 10 x^{5} + 2 x^{3} - x

is an even or odd function.

$10 x^{5}$ ‍ is an odd function, since it's a monomial of $odd$ ‍ degree.
$2 x^{3}$ ‍ is an odd function, since it's a monomial of $odd$ ‍ degree.
$- x$ ‍ is an odd function since it is a monomial of $odd$ ‍ degree. Notice it can be written as $- 1 \cdot x^{1}$ ‍.

Since each term in the polynomial function is itself an odd function (odd degree), the polynomial is also odd.

We could also see this algebraically by showing that

h (- x) = - h (x) .

\begin{aligned} h (- x) & = 10 (- x)^{5} + 2 (- x)^{3} - (- x) \\ = 10 (- x^{5}) + 2 (- x^{3}) - (- x) & (- x)^{n} = - x^{n} when n is odd \\ = - 10 x^{5} - 2 x^{3} + x & Simplify \\ = - h (x) \end{aligned}

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Algebra (all content)

Course: Algebra (all content) > Unit 10

What you should be familiar with before taking this lesson

What you will learn in this lesson

Investigation: Symmetry of monomials

Concluding the investigation

Investigation: Symmetry of polynomials

Example 1: f(x)=2x4−3x2−5‍

Example 2: g(x)=5x7−3x3+x‍

Example 3: h(x)=2x4−7x3‍

Concluding the investigation

Check your understanding

Want to join the conversation?

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

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

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