Algebraic Invariance Examples in Math

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

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

Concept Recap

Algebraic properties or quantities that remain unchanged when specific algebraic transformations are applied to an expression or system.

The degree of a polynomial doesn't change when you multiply it by a non-zero constant.

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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 algebraic invariant is a quantity that stays the same after an allowed transformation.

Common stuck point: The procedure for algebraic invariance is the easy part; the trap is assuming all features are invariant under a transformation. Asking "Does this quantity stay exactly the same after the allowed transformation is applied?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this quantity stay exactly the same after the allowed transformation is applied?

Worked Examples

Example 1

medium

The polynomial

2x^3 + 5x^2 - x + 3

can be rewritten as

2(x+1)^3 + (x+1)^2 - 4(x+1) + 5

. What is invariant?

Answer

Degree (3) and leading coefficient (2) are invariant.

First step

Step 1: The degree is 3 in both forms — degree is invariant.

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Example 2

hard

Show that the discriminant

b^2 - 4ac

is invariant under the substitution

x = t + k

ax^2 + bx + c = 0

Example 3

medium

Under the substitution

x \to x + 3

, does the discriminant of

x^2 - 5x + 6

change?

Example 4

medium

Under the substitution

u = x + 1

, transform

f(x) = x^2 + 2x + 3

. What stays the same?

Example 5

hard

x_{n+1} = x_n + y_n

y_{n+1} = x_n - y_n

starts with

x_0 = 3, y_0 = 1

. Show

x_n^2 - 2 y_{n+1}^2 \cdot ?

— find

x_n^2 + y_n^2

at step 1.

Example 6

challenge

Given a triple

(a, b, c)

on which the move

(a, b, c) \to (a+b, b, c-b)

is applied repeatedly. What sum is invariant?

Practice Problems

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

Example 1

easy

When you factor

x^2 + 5x + 6 = (x+2)(x+3)

, what is invariant?

Example 2

medium

What changes and what stays the same when multiplying both sides of

2x = 6

by 3?

Example 3

easy

Does multiplying the polynomial

x^2+3x+1

by 5 change its degree?

Example 4

easy

Under the substitution

x\to x+1

, does the degree of

x^2

stay the same? Compute the new polynomial.

Example 5

easy

For

2x^2 - 4x + 6

, the sum of roots is

-b/a

. Does dividing the whole equation by 2 change the sum of roots?

Example 6

easy

Is the value of

x^2+y^2

invariant when you swap

x

and

y

Example 7

easy

Does adding 3 to both sides of

x-5=2

change its solution?

Example 8

easy

Is the determinant of a matrix invariant under transposition? (Is

\det(A)=\det(A^T)

Example 9

easy

Under reflection

x\to -x

, is

x^2

invariant? Evaluate

(-x)^2

Example 10

easy

Does the number of solutions of

x^2=4

change if you write it as

x^2-4=0

Example 11

medium

Under scaling

f(x)\to 3f(x)

, which is invariant: the roots of

f

or the

y

-intercept?

Example 12

medium

The discriminant

b^2-4ac

x^2-5x+6

is 1. If we shift roots by replacing

x

with

x-2

, is the discriminant invariant? Check.

Example 13

medium

Is the sum

x+y

invariant under the transformation

(x,y)\to(x+3,y-3)

Example 14

medium

Which is invariant when a quadratic

ax^2+bx+c

is multiplied by a nonzero constant

k

: the roots, or the coefficients?

Example 15

medium

Find an invariant of the rotation-like map

(x,y)\to(-y,x)

. Test

x^2+y^2

Example 16

medium

Is the product of roots

c/a

3x^2+12x+9

invariant when the equation is divided by 3?

Example 17

medium

A student claims 'the value of

x^2+1

is invariant under any substitution'. Disprove with a specific substitution.

Example 18

medium

Is the parity (even/odd) of

n^2+n

invariant for all integers

n

? Determine which parity.

Example 19

medium

Is the area of a triangle invariant when its base and height are swapped in

A=\frac12 bh

Example 20

challenge

Show that the difference of the roots' squares' relation, specifically

b^2-4ac

scaled, behaves how under

a,b,c\to ka,kb,kc

? Determine the scaling factor.

Example 21

challenge

Prove that the trace of a

2\times2

matrix is invariant under the similarity transform

A\to P^{-1}AP

(state the key property used).

Example 22

challenge

The expression

\frac{x-y}{x-z}

is part of the cross-ratio. Find what stays invariant when

x,y,z

all shift by a constant

t

(x,y,z)\to(x+t,y+t,z+t)

Example 23

easy

Is the value

x + y

invariant under the swap

x \leftrightarrow y

Example 24

easy

x - y

invariant under the swap

x \leftrightarrow y

Example 25

easy

Adding 5 to both sides of

x = 7

, does the solution change?

Example 26

easy

Is the GCD of

a, b

invariant under swapping

a

and

b

Example 27

easy

If you multiply both sides of

2x + 4 = 10

by 3, is the solution invariant?

Example 28

medium

Under the transformation

(x, y) \to (x + 2, y - 2)

, is

x + y

invariant?

Example 29

medium

Is the product of the roots of

2x^2 - 8x + 6

invariant when we divide the equation by 2?

Example 30

medium

Under rotation

(x, y) \to (y, -x)

, is

xy

invariant?

Example 31

medium

Under

(x,y) \to (y, -x)

, is

x^2 + y^2

invariant?

Example 32

medium

Is the parity of

n^3 - n

invariant for all integers

n

? Determine which parity.

Example 33

medium

n^2 \bmod 3

invariant for all integers

n \not\equiv 0 \pmod 3

? Determine the value.

Example 34

medium

An expression in

x

is invariant under

x \to x + 1

and equals

5

x = 0

. What is its value at

x = 100

Example 35

hard

Show that the determinant of a

2 \times 2

matrix is invariant under transpose.

Example 36

hard

If we scale

(a, b, c) \to (3a, 3b, 3c)

ax^2 + bx + c = 0

, do the roots change?

Example 37

hard

Under the substitution

x \to 1/x

, is

f(x) = x + 1/x

invariant (for

x \ne 0

Example 38

hard

On a 4x4 chessboard, two opposite corners are removed. Can the remaining 14 squares be tiled by 1x2 dominoes? Give the invariant.

Example 39

hard

A sequence is defined by

a_{n+1} = 3a_n - 2

. If

a_0 = 5

, what is the invariant 'shift' that makes the recurrence purely multiplicative?

Example 40

challenge

Show that the trace of a

2 \times 2

matrix is invariant under conjugation

A \to P^{-1} A P

Example 41

challenge

A pile starts with

N

stones. At each step a move takes one pile and either doubles it or removes 1 stone. After many moves can the parity invariant be used to predict residue mod 3? Start

N=2

, can we reach

N=10

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Related Concepts

Expressions Invariants Symmetric Functions

Background Knowledge

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

expressions