Question

please answer the question

Answer 1

Question :-

When a polynomial 2x³ + 3x² + ax + b is divided by (x - 2) leaves remaider 2, and (x + 2) leaves remainder - 2. Find a and b

Answer :-

Value of a is - 7 and b is - 12

Solution :-

Let f(x) = 2x³ + 3x² + ax + b

(i) when f(x) divided by (x - 2) leaves remaider 2

First find zero of x - 2

To find zero equate (x - 2) to 0

x - 2 = 0

x = 2

By remainder theorem f(2) is the remainder

Given

f(2) = 2

⇒ 2(2)³ + 3(2)² + a(2) + b = 2

⇒ 2(8) + 3(4) + 2a + b = 2

⇒ 16 + 12 + 2a + b = 2

⇒ 28 + 2a + b = 2

⇒ b = 2 - 28 - 2a

⇒ b = - 26 - 2a ---(1)

(ii) When f(x) divided by (x + 2) leaves remaider - 2

First find zero of x + 2

To find zero equate (x + 2) to 0

x + 2 = 0

x = - 2

By remainder theorem f(-2) is the remainder

Given

f(-2) = -2

⇒ 2(-2)³ + 3(-2)² + a(-2) + b = - 2

⇒ 2(-8) + 3(4) - 2a + b = - 2

⇒ - 16 + 12 - 2a + b = - 2

⇒ - 4 - 2a + b = - 2

⇒ b = - 2 + 4 + 2a

⇒ b = 2 + 2a ---(2)

From (1) & (2)

- 26 - 2a = 2 + 2a

⇒ - 26 - 2 = 2a + 2a

⇒ - 28 = 4a

⇒ - 28/4 = a

⇒ - 7 = a

⇒ a = - 7

Substitute a = - 7 in (1)

b = - 26 - 2(-7)

⇒ b = - 26 + 14

⇒ b = - 12

Therefore value of a is - 7 and b is - 12