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.

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: Invariants help identify what's essential vs. what can change.

Common stuck point: Carefully verifying which properties are truly invariant under the specific transformation requires checking both directions.

Sense of Study hint: Transform the expression (expand, factor, substitute) and check which quantities stayed the same.

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?

Solution

  1. 1
    Step 1: The degree is 3 in both forms — degree is invariant.
  2. 2
    Step 2: The leading coefficient is 2 in both — also invariant.
  3. 3
    Step 3: The specific coefficients of each power change, but the polynomial's behavior (degree, leading term) doesn't.

Answer

Degree (3) and leading coefficient (2) are invariant.
An algebraic invariant is a property that doesn't change when an expression is rewritten in equivalent forms. Degree and leading coefficient are invariant under variable substitution.

Example 2

hard
Show that the discriminant b^2 - 4ac is invariant under the substitution x = t + k in ax^2 + bx + c = 0.

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?
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Background Knowledge

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

expressions