Practice Algebraic Symmetry in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

x + y = 10

and

xy = 21

, find

x^2 + y^2

Example 2

easy

Is the expression

x + y

symmetric in

x

and

y

Example 3

medium

x + y = 7

and

xy = 12

, find

x

and

y

Example 4

medium

Determine whether

x^2 + 3xy + y^2

is symmetric in

x,y

Example 5

hard

x + y = 3

and

xy = 2

, find

x^4 + y^4

Example 6

easy

x^2 - y^2

symmetric in

x

and

y

Example 7

challenge

x + y = 4

and

x^3 + y^3 = 28

, find

xy

Example 8

hard

x, y

satisfy

x + y = 5

and

xy = 6

, what is

(x - y)^2

and

|x - y|

Example 9

hard

x + y = 4

and

x^2 + y^2 = 10

, find

xy

and

x^4 + y^4

Example 10

medium

Use symmetry to compute

x^2+y^2

given

x+y=5

and

xy=6

Example 11

medium

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

fully symmetric in

x, y, z

Example 12

hard

Given

x + y + z = 6

xy + yz + zx = 11

, find

x^2 + y^2 + z^2

Example 13

easy

(x + y)^2

symmetric?

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

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 16

easy

x^3 + y^3

symmetric in

x

and

y

Example 17

hard

For

x + y = s

xy = p

, express

(x - y)^2

in terms of

s

and

p

Example 18

medium

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

symmetric in

x

and

y

Example 19

medium

(x - y)^2

symmetric in

x

and

y

Example 20

easy

x^2 + y^2

symmetric in

x

and

y

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Related Concepts

Expressions Symmetric Functions Invariants