Question

The polynomial p(x) =x^4-2x^3+3x^2-ax+b when divided by (x-1) and (x+1) leaves the remainder 5 and 19 respectively. Find the value of a and b. Hence, find the remainder when p(x) is divided by (x-2)​

Answer 1

Answer:

Given :-

f(x) = x⁴ - 2x³+ 3x² - ax + b

To Find :-

Value of a and b.

Solution :-

f(x) = x⁴ - 2x³+ 3x² - ax + b

According to Question,

When f(x) is divided by (x-1), it leaves a remainder 5

⇒ f(1) = 5

⇒ 1 - 2(1)³+ 3(1)² - a(1) + b = 5

⇒ 1 - 2 + 3 - a +b = 5

⇒ -a + b = 3 … (i)

When f(x) is divided by (x+1), it leaves a remainder 19

⇒ f(-1) = 19

⇒ (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b = 19

⇒ 1 + 2 + 3 + a + b = 19

⇒ a +b = 13 … (ii)

Adding (i) and (ii), we get

⇒ 2b = 16

⇒ b = 16/2

⇒ b = 8

⇒ a = b - 3

⇒ a = 8 - 3

⇒ a = 5

Hence, the value of a and b are 5 and 8.