Exercises on GCD and LCM of Polynomials

Exercise 1: Finding the GCD of Polynomials

Find the greatest common divisor (GCD) of the following polynomials:

P(x) = x³ + 2x² − x − 2 and Q(x) = x² + x − 2

Step 1: Factorize both polynomials.

P(x) = (x + 2)(x - 1)(x + 1)

Q(x) = (x + 2)(x - 1)

Step 2: Identify the common factors.

GCD(P, Q) = (x + 2)(x - 1)

Exercise 2: Finding the LCM of Polynomials

Find the least common multiple (LCM) of the following polynomials:

P(x) = x² − 4x + 4 and Q(x) = x² − 2x

Step 1: Factorize both polynomials.

P(x) = (x - 2)(x - 2)

Q(x) = x(x - 2)

Step 2: Compute the LCM.

LCM(P, Q) = x(x - 2)(x - 2)

Exercise 3: Application Problems

Find the GCD for the given problem:

A(x) = x³ − 4x² + 3x and B(x) = x² − 3x

Step 1: Factorize the given expressions.

A(x) = x(x - 1)(x - 3)

B(x) = x(x - 3)

Step 2: Identify the GCD.

GCD(A, B) = x(x - 3)