Question

Find the LCM of x^3 – x^2 – 2x and x^4 – 4x^2. Show your work.

x^3(x + 2)(x – 2)^2(x + 1)

x(x – 2)(x + 2)(x + 1)

x^2(x – 2)(x + 2)(x + 1)

Answer 1

Start by factoring each polynomial.

x^3 − x^2 − 2x = x(x^2 − x − 2) = x(x − 2)(x + 1).

x^4 − 4x^2 = x^2(x^2 − 4) = x^2(x − 2)(x + 2).

The LCM takes each distinct factor to the highest power it appears in either polynomial:
- x appears to power 2 (max of 1 and 2) → x^2
- (x − 2) appears to power 1 → (x − 2)
- (x + 1) appears to power 1 → (x + 1)
- (x + 2) appears to power 1 → (x + 2)

Thus LCM = x^2(x − 2)(x + 2)(x + 1), which is the third option.