If (x + 1) and (x – 1) are factors of the polynomial x⁴ + ax³ – 3x² + 2x + b then find the value of a and b.

Using Remainder Theorem:

Putting first factor = 0, we get -

(x + 1) = 0

⇒ x = – 1.

Now, ∵ it is given that (x + 1) is a factor of the polynomial x⁴ + ax³ – 3x² + 2x + b

⇒ If we substitute x = – 1 in x⁴ + ax³ – 3x² + 2x + b, the remainder will be zero.

⇒ (– 1)⁴ + a(– 1)³ – 3(– 1)² + 2(– 1) + b = 0

⇒ 1 – a – 3 – 2 + b = 0

⇒ – a + b – 4 = 0 ……(eqn 1)

Also, Using Remainder Theorem:

Putting second factor = 0, we get -

(x – 1) = 0

⇒ x = 1.

Now, ∵ it is given that (x – 1) is a factor of the polynomial x⁴ + ax³ – 3x² + 2x + b

⇒ If we substitute x = 1 in x⁴ + ax³ – 3x² + 2x + b, the remainder will be zero.

⇒ (1)⁴ + a(1)³ – 3(1)² + 2(1) + b = 0

⇒ 1 + a – 3 + 2 + b = 0

⇒ a + b = 0 ……(eqn 2)

Now, adding eqn1 and eqn 2, we get -

– a + b – 4 + a + b = 0

⇒ 2b – 4 = 0

⇒ b = 2

Now, substituting b = 2 in eqn2. We get -

a + 2 = 0

⇒ a = – 2