Algebraic Symmetry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Algebraic Symmetry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The property of an expression or equation that remains unchanged when certain transformations — such as swapping variables — are applied.

$x^2 + y^2$ is symmetric: swapping $x$ and $y$ gives the same expression.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An expression is symmetric if exchanging its variables leaves it identical.

Common stuck point: The procedure for algebraic symmetry is the easy part; the trap is assuming any expression with both variables is symmetric. Asking "If I swap the two variables, do I get back the exact same expression?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: If I swap the two variables, do I get back the exact same expression?

Worked Examples

Example 1

easy

f(x, y) = x^2 + y^2

symmetric in

x

and

y

Answer

Yes,

f

is symmetric.

First step

Step 1: Check if

f(x, y) = f(y, x)

Full solution

2
Step 2: $f(y, x) = y^2 + x^2 = x^2 + y^2 = f(x, y)$ .
3
Step 3: Yes, it is symmetric — swapping $x$ and $y$ doesn't change the expression.

An expression is symmetric in

x

and

y

if swapping them produces the same expression. This symmetry often simplifies problem-solving — if

(a, b)

is a solution, so is

(b, a)

Example 2

medium

f(x, y) = x^2 - xy + y^2

symmetric?

Example 3

medium

Given

x + y = 6

and

xy = 5

, compute

x^2 + y^2

Example 4

hard

x + y = 4

and

x^2 + y^2 = 10

, find

xy

and

x^4 + y^4

Example 5

hard

x, y

satisfy

x + y = 5

and

xy = 6

, what is

(x - y)^2

and

|x - y|

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

f(x, y) = x - y

symmetric?

Example 2

medium

x + y = 10

and

xy = 21

, find

x^2 + y^2

Example 3

easy

Is the expression

x + y

symmetric in

x

and

y

Example 4

easy

x^2 + y^2

symmetric in

x

and

y

Example 5

easy

x^2 + xy

symmetric in

x

and

y

Example 6

easy

xy

symmetric in

x

and

y

Example 7

easy

Is the equation

x = y

symmetric in

x

and

y

Example 8

easy

x - y

symmetric in

x

and

y

Example 9

easy

x^2 + y^2 + z^2

symmetric in

x

y

, and

z

Example 10

easy

Which is symmetric in

x

and

y

: (A)

x^2 y

or (B)

x^2 y + x y^2

Example 11

medium

Use symmetry to compute

x^2+y^2

given

x+y=5

and

xy=6

Example 12

medium

f(x,y)=\frac{x-y}{x+y}

symmetric, antisymmetric, or neither?

Example 13

medium

Determine whether

x^2 + 3xy + y^2

is symmetric in

x,y

Example 14

medium

If a two-variable polynomial

P(x,y)

is symmetric and

P(2,3)=7

, what is

P(3,2)

Example 15

medium

Compute

x^3+y^3

given

x+y=4

and

xy=3

Example 16

medium

Is the system

\{x+y=6,\ xy=8\}

symmetric in

x

and

y

? What does that imply about its solutions?

Example 17

medium

Verify that

|x-y|

is symmetric in

x

and

y

Example 18

medium

Rewrite the symmetric expression

x^2y+xy^2

in terms of

s=x+y

and

p=xy

Example 19

medium

\frac{1}{x}+\frac{1}{y}

symmetric in

x

and

y

? Express it via

s=x+y

p=xy

Example 20

challenge

Show that any symmetric polynomial in

x,y

that is also a function of

x+y

alone must be independent of

xy

, then test

x^2+y^2

Example 21

challenge

For which

k

x^2 + kxy + y^2

a perfect square trinomial, and is it still symmetric for that

k

Example 22

challenge

Prove that the discriminant condition for

x+y=s

xy=p

to have real solutions is

s^2\ge 4p

, and explain why this uses only symmetric data.

Example 23

easy

x^3 + y^3

symmetric in

x

and

y

Example 24

easy

x^2 y

symmetric in

x

and

y

Example 25

easy

Is the function

f(x, y) = y - x

symmetric, antisymmetric, or neither?

Example 26

easy

Is the product

xy

symmetric in

x

and

y

Example 27

easy

x^2 - y^2

symmetric in

x

and

y

Example 28

medium

Given

x + y = 5

and

xy = 4

, compute

x^3 + y^3

Example 29

medium

P(x, y)

is symmetric and

P(5, -1) = 12

, find

P(-1, 5)

Example 30

medium

f(x, y, z) = xy + yz + zx

fully symmetric in

x, y, z

Example 31

medium

Rewrite

x^2 + y^2

in terms of

s = x + y

and

p = xy

Example 32

medium

For

x + y = 8

and

xy = 12

, compute

\dfrac{1}{x} + \dfrac{1}{y}

Example 33

medium

x + y = 7

and

xy = 12

, find

x

and

y

Example 34

medium

(x - y)^2

symmetric in

x

and

y

Example 35

hard

Express

x^3 - y^3

in terms of

x - y

and

xy

, given

x + y = s

Example 36

hard

Given

x + y + z = 6

xy + yz + zx = 11

, find

x^2 + y^2 + z^2

Example 37

hard

x + y = 3

and

xy = 2

, find

x^4 + y^4

Example 38

hard

For

x + y = s

xy = p

, express

(x - y)^2

in terms of

s

and

p

Example 39

medium

Is the system

\{x + y = 10,\ x - y = 4\}

symmetric in

x

and

y

Example 40

medium

Given

x + y + z = 0

, prove

x^3 + y^3 + z^3 = 3xyz

Example 41

challenge

x + y = 4

and

x^3 + y^3 = 28

, find

xy

Example 42

challenge

Let

s_k = x^k + y^k

. If

x + y = 2

and

xy = -1

, find

s_4

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Related Concepts

Expressions Symmetric Functions Invariants

Background Knowledge

These ideas may be useful before you work through the harder examples.

expressions