Algebraic Invariance Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
The polynomial 2x3+5x2−x+32x^3 + 5x^2 - x + 3 can be rewritten as 2(x+1)3+(x+1)2−4(x+1)+52(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.

About Algebraic Invariance

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

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More Algebraic Invariance Examples

Example 2 hard

Show that the discriminant [formula] is invariant under the substitution [formula] in [formula].

Example 3 easy

When you factor [formula], what is invariant?

Example 4 medium

What changes and what stays the same when multiplying both sides of [formula] by 3?

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