Question

If f(x) = x4 – 2x3 + 3x2 – ax + b is divided by (x-1)
and (x + 1) it leaves the remainders 5 and 19 respectively,
Find a and b​

Answer 1

Answer:

a = 5 and b = 8

Step-by-step explanation:

Solution;

Given;

f(x) = x4 - 2x3 +3x2 - ax + b

Case I,

x - a = x - 1

or, x = 1

Remainder (R) = 5

or, f(a) = 5

or, f(1) = 5

or, (1)4 - 2(1)3 +3(1)2 - a(1) + b = 5

or, 1 - 2 + 3 - a + b - 5 = 0

or, 4 - 7 + b = a

or, b - 3 = a

or, a = b - 3.............(1)

Case II,

x - a = x + 1

or, a = -1

Remainder (R) = 19

or, f(a) = 19

or, f(-1) = 19

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

or, 1 - 2(-1) + 3(1) + a + b - 19 = 0

or, 1 +2 + 3 - 19 + b - 3 + b = 0 [From (i)]

or, 6 - 22 + 2b = 0

or, -16 + 2b = 0

or, 2b = 16

or b = 16/2

or, b = 8

Finally,

Substituting the value of b in (i)

a = b - 3

= 8 - 3

= 5

Hence, the required values of a and b are 5 and 8 respectively.