# 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, left parenthesis, minus, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis 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, left parenthesis, minus, x, right parenthesis, equals, minus, f, left parenthesis, x, right parenthesis 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, left parenthesis, x, right parenthesis, equals, a, x, start superscript, n, end superscript 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, left parenthesis, minus, x, right parenthesis:
• If f, left parenthesis, minus, x, right parenthesis is the same as f, left parenthesis, x, right parenthesis, then we know f is even.
• If f, left parenthesis, minus, x, right parenthesis is the opposite of f, left parenthesis, x, right parenthesis, then we know f is odd.
• Otherwise, it is neither even nor odd.
As a first example, let's determine whether f, left parenthesis, x, right parenthesis, equals, 4, x, start superscript, 3, end superscript is even, odd, or neither.
Here f, left parenthesis, minus, x, right parenthesis, equals, minus, f, left parenthesis, x, right parenthesis, and so function f is an odd function.
The graph of y, equals, f, left parenthesis, x, right parenthesis is symmetric with respect to the origin, which confirms our solution!
Now try some examples on your own to see if you can find a pattern.
1) Is g, left parenthesis, x, right parenthesis, equals, 3, x, start superscript, 2, end superscript even, odd, or neither?

To determine whether g is even, odd, or neither, let's find g, left parenthesis, minus, x, right parenthesis.
Since g, left parenthesis, minus, x, right parenthesis, equals, g, left parenthesis, x, right parenthesis, the function is an even function.
2) Is h, left parenthesis, x, right parenthesis, equals, minus, 2, x, start superscript, 5, end superscript even, odd, or neither?

To determine whether h is even, odd, or neither, let's find h, left parenthesis, minus, x, right parenthesis.
Since h, left parenthesis, minus, x, right parenthesis, equals, minus, h, left parenthesis, x, right parenthesis, the function is an odd function.

### 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 FunctionOdd Function
Examples g, left parenthesis, x, right parenthesis, equals, 3, x, start superscript, start color purpleC, 2, end color purpleC, end superscripth, left parenthesis, x, right parenthesis, equals, minus, 2, x, start superscript, start color greenD, 5, end color greenD, end superscript
In generalf, left parenthesis, x, right parenthesis, equals, a, x, start superscript, start color purpleC, n, end color purpleC, end superscript where n is start color purpleC, e, v, e, n, end color purpleCf, left parenthesis, x, right parenthesis, equals, a, x, start superscript, start color greenD, n, end color greenD, end superscript where n is start color greenD, o, d, d, end color greenD

This is because left parenthesis, minus, x, right parenthesis, start superscript, n, end superscript, equals, x, start superscript, n, end superscript when n is even and left parenthesis, minus, x, right parenthesis, start superscript, n, end superscript, equals, minus, x, start superscript, n, end superscript 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: $f(x)=2x^4-3x^2-5$f, left parenthesis, x, right parenthesis, equals, 2, x, start superscript, 4, end superscript, minus, 3, x, start superscript, 2, end superscript, minus, 5

To determine whether f is even, odd, or neither, we find f, left parenthesis, minus, x, right parenthesis.
Since f, left parenthesis, minus, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, function f is an even function.
Note that all the terms of f are of an even degree.

### Example 2: $g(x)=5x^7-3x^3+x$g, left parenthesis, x, right parenthesis, equals, 5, x, start superscript, 7, end superscript, minus, 3, x, start superscript, 3, end superscript, plus, x

Again, we start by finding g, left parenthesis, minus, x, right parenthesis.
At this point, notice that each term in g, left parenthesis, minus, x, right parenthesis is the opposite of each term in g, left parenthesis, x, right parenthesis. In other words, g, left parenthesis, minus, x, right parenthesis, equals, minus, g, left parenthesis, x, right parenthesis, and so g is an odd function.
Note that all the terms of g are of an odd degree.

### Example 3: $h(x)=2x^4-7x^3$h, left parenthesis, x, right parenthesis, equals, 2, x, start superscript, 4, end superscript, minus, 7, x, start superscript, 3, end superscript

Let's find h, left parenthesis, minus, x, right parenthesis.
\begin{aligned}h(\blueD{-x})&=2(\blueD{-x})^4-7(\blueD{-x})^3\\ \\ &=2(x^4)-7(-x^3)&&\small{\gray{(-x)^4=x^4\text{ and } (-x)^3=-x^3}}\\ \\ &=2x^4+7x^3&&\small{\gray{\text{Simplify}}}\\\\ \end{aligned}
2, x, start superscript, 4, end superscript, plus, 7, x, start superscript, 3, end superscript is not the same as h, left parenthesis, x, right parenthesis nor is it the opposite of h, left parenthesis, x, right parenthesis.
Mathematically, h, left parenthesis, minus, x, right parenthesis, does not equal, h, left parenthesis, x, right parenthesis and h, left parenthesis, minus, x, right parenthesis, does not equal, minus, h, left parenthesis, x, right parenthesis, and so h is neither even nor odd.
Note that hhas 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.
empty spaceGeneral ruleExample polynomial
EvenA polynomial is even if each term is an even function.f, left parenthesis, x, right parenthesis, equals, 2, x, start superscript, 4, end superscript, minus, 3, x, start superscript, 2, end superscript, minus, 5
OddA polynomial is odd if each term is an odd function.g, left parenthesis, x, right parenthesis, equals, 5, x, start superscript, 7, end superscript, minus, 3, x, start superscript, 3, end superscript, plus, x
NeitherA polynomial is neither even nor odd if it is made up of both even and odd functions.h, left parenthesis, x, right parenthesis, equals, 2, x, start superscript, 4, end superscript, minus, 7, x, start superscript, 3, end superscript

### Check your understanding

3) Is f, left parenthesis, x, right parenthesis, equals, minus, 3, x, start superscript, 4, end superscript, minus, 7, x, start superscript, 2, end superscript, plus, 5 even, odd, or neither?

To start, let's decide if each term of f, left parenthesis, x, right parenthesis, equals, minus, 3, x, start superscript, 4, end superscript, minus, 7, x, start superscript, 2, end superscript, plus, 5 is an even or odd function.
• minus, 3, x, start superscript, start color purpleC, 4, end color purpleC, end superscript is an even function, since it's a monomial of start color purpleC, e, v, e, n, end color purpleC degree.
• minus, 7, x, start superscript, start color purpleC, 2, end color purpleC, end superscript is an even function, since it's a monomial of start color purpleC, e, v, e, n, end color purpleC degree.
• 5 is an even function since it is a monomial of start color purpleC, e, v, e, n, end color purpleC degree. Notice it can be written as 5, dot, x, start superscript, start color purpleC, 0, end color purpleC, end superscript.
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, left parenthesis, minus, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, point
\begin{aligned} &\phantom{=}f(\blueD{-x}) \\\\ &=-3(\blueD{-x})^4-7(\blueD{-x})^2+5 \\\\ &=-3(x^4)-7(x^2)+5&&\small{\gray{(-x)^n=x^n\text{ for even }n}} \\\\ &=-3x^4-7x^2+5&&\small{\gray{\text{Simplify}}} \\\\ &=f(x) \end{aligned}
4) Is g, left parenthesis, x, right parenthesis, equals, 8, x, start superscript, 7, end superscript, minus, 6, x, start superscript, 3, end superscript, plus, x, start superscript, 2, end superscript even, odd, or neither?

To start, let's decide if each term of g, left parenthesis, x, right parenthesis, equals, 8, x, start superscript, 7, end superscript, minus, 6, x, start superscript, 3, end superscript, plus, x, start superscript, 2, end superscript is an even or odd function.
• 8, x, start superscript, start color greenD, 7, end color greenD, end superscript is an odd function, since it's a monomial of start color greenD, o, d, d, end color greenD degree.
• minus, 6, x, start superscript, start color greenD, 3, end color greenD, end superscript is an odd function, since it's a monomial of start color greenD, o, d, d, end color greenD degree.
• x, start superscript, 2, end superscript is an even function, since it is a monomial of start color purpleC, e, v, e, n, end color purpleC 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, left parenthesis, minus, x, right parenthesis is neither g, left parenthesis, x, right parenthesis nor its opposite.
\begin{aligned}g(\blueD{-x})&=8(\blueD{-x})^7-6(\blueD{-x})^3+(\blueD{-x})^2\\ \\ &=8(-x^7)-6(-x^3)+x^2\\ \\ &=-8x^7+6x^3+x^2&&\small{\gray{\text{Simplify}}}\\\\ \end{aligned}
5) Is h, left parenthesis, x, right parenthesis, equals, 10, x, start superscript, 5, end superscript, plus, 2, x, start superscript, 3, end superscript, minus, x even, odd, or neither?