🎨 GCD and LCM of Polynomials

🔍 Understanding GCD and LCM

The Greatest Common Divisor (GCD) of two polynomials is the highest degree polynomial that divides both without a remainder.

The Least Common Multiple (LCM) of two polynomials is the lowest degree polynomial that is a multiple of both.

Step 1: Factor each polynomial completely.

Step 2: Identify the common factors for GCD.

Step 3: Multiply the lowest powers of common factors to find GCD.

Step 4: Identify all factors present in either polynomial for LCM.

Step 5: Multiply the highest powers of all factors to find LCM.

Example: Find the GCD and LCM of:

Given polynomials: \( f(x) = x^3 - 3x^2 + 3x - 1 \), \( g(x) = x^2 - 2x + 1 \).

Factorizing: \( f(x) = (x - 1)^3 \), \( g(x) = (x - 1)^2 \).

GCD: Multiply lowest powers of common factors → \( (x - 1)^2 \).

LCM: Multiply highest powers of all factors → \( (x - 1)^3 \).