Question

If the polynomial x³ + ax² − bx − 6 is exactly divisible by (x² − x − 2), find the values of a and b.

Answer 1

Let f(x) = x³ + ax² − bx − 6 and g(x) = x² − x − 2 be the given polynomials. 

The divisor polynomial is x² − x − 2, which can be factored into its linear factors:

x² − x − 2 = (x − 2) (x + 1)

We substitute x = 2 and x = −1 into f(x) to form two equations:

f(2) = 0

⇒ 2³ + a(2)² − b(2) − 6 = 0

⇒ 8 + 4a − 2b − 6 = 0

⇒ 4a − 2b + 2 = 0

⇒ 2a − b + 1 = 0   ...(i)

f(−1) = 0

⇒ (−1)³ + a(−1)² − b(−1) − 6 = 0

⇒ −1 + a + b − 6 = 0

⇒ a + b − 7 = 0   ...(ii)

Add (i) from (ii),

(2a − b + 1) + (a + b − 7) = 0

3a − 6 = 0

3a = 6

a = `6/3`

a = 2

Now substitute a = 2 in equation (ii)

a + b − 7 = 0

2 + b − 7 = 0

b − 5 = 0

b = 5

The values are a = 2 and b = 5.