Question

If x + 1 and x - 1 are factors of ax3 + x2 - 2x + b, then the values of a and b are respectively
(1) 2 - 1
(2) - 1, 2
(3) 1,2
(4) -1, -2​

Answer 1

Given :

• Polynomial = ax³ + x² - 2x + b
• The factors of the polynomial = (x + 1) and (x - 1)

To find :

• The value of a and b

Solution :

⇒ P(x) = ax³ + x² - 2x + b

The factor of the given polynomial = (x + 1)

→ (x + 1) = 0

→ x = - 1

⇒ P(-1) = a(-1)³ + (-1)² - 2(-1) + b = 0

⇒ a(-1) + (1) + 2 + b = 0

⇒ - a + 1 + 2 + b = 0

⇒ - a + b + 3 = 0

⇒ - a + b = - 3 -------(1)

⇒ P(x) = ax³ + x² - 2x + b

The factor of the given polynomial = (x - 1)

→ (x - 1) = 0

→ x = 1

⇒ P(1) = a(1)³ + (1)² - 2(1) + b = 0

⇒ a(1) + (1) - 2 + b = 0

⇒ a + 1 - 2 + b = 0

⇒ a + b - 1 = 0

⇒ a + b = 1 -------(2)

Solving (1) and (2) :-

⠀⠀⠀⠀⠀⠀⠀⠀⠀ - a + b = - 3

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀a + b = 1

⠀⠀⠀⠀⠀⠀⠀⠀⠀____________

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀2b = - 2

⠀⠀⠀⠀⠀⠀⠀⠀⠀____________

⇒ 2b = - 2

⇒ b = - 1

Substituting the value of 'b' in equation '1' :-

⇒ - a + b = - 3

⇒ - a + (-1) = - 3

⇒ - a - 1 = - 3

⇒ - a = - 3 + 1

⇒ - a = - 2

⇒ a = 2

Therefore, the value of b = - 1 and a = 2

Answer → Option (2) - 1, 2