Algebraic Invariance Math Example 3

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Example 3

easy
When you factor x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3), what is invariant?

Solution

  1. 1
    The value of the expression for any xx is invariant — both forms give the same output.
  2. 2
    Also: the roots (x=−2,−3x = -2, -3), the degree (2), and the leading coefficient (1).

Answer

The polynomial itself (same values, roots, degree).
Factoring is an equivalent rewriting — it changes the form but not the function. The polynomial's value at every point, its roots, and its graph are all invariant under factoring.

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 1 medium

The polynomial [formula] can be rewritten as [formula]. What is invariant?

Example 2 hard

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

Example 4 medium

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

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