Factorise 𝑥ଷ − 2𝑥ଶ − 𝑥 + 2 using factor theorem

Sure, let's factorize the given polynomial step by step using the factor theorem.

The given polynomial is 𝑥³ − 2𝑥² − 𝑥 + 2.

Step 1: Find the roots of the polynomial by setting it equal to zero and solving for x.

x³ - 2x² - x + 2 = 0

We can try some integer values of x to see if they are roots.

Let's try x = 1:

(1)³ - 2(1)² - 1 + 2 = 1 - 2 - 1 + 2 = 0

So, x = 1 is a root of the polynomial.

Step 2: Divide the polynomial by (x - 1) using synthetic division or long division.

The result is x² - x - 2.

Step 3: Factorize the quadratic polynomial.

x² - x - 2 = 0

This can be factored into (x - 2)(x + 1) = 0

So, the roots of the polynomial are x = 2 and x = -1.

Step 4: Write the original polynomial as a product of linear factors.

So, 𝑥³ − 2𝑥² − 𝑥 + 2 = (x - 1)(x - 2)(x + 1)

And that's the factorization of the given polynomial using the factor theorem.