Question

x3 + x + 2 factorise

Answer 1

Answer:

x³ + x + 2 = ( x + 1 ) ( x² - x + 2 )

Step-by-step explanation:

To factorize f(x) = x³ + x + 2, we want to "see" at least one factor.

The Factor Theorem tells us that (x-a) is a factor if f(a) = 0.  So look for such a value a.

Notice f(-1) = (-1)³ + (-1) + 2 = -1 - 1 + 2 = 0.  It follows that (x+1) is a factor.

Now do the division:

x³ + x + 2 = ( x + 1 ) ( x² - x + 2 )

The discriminant of the quadratic factor is (-1)² - 4×2 = 1-8 = -7.  This is not a square, so the quadratic doesn't factorize any further.