Question

please try to answer it.............

Answer 1

GIVEN
X^3 - 10x^2 + ax + b is divisible by (x - 1) & (x - 2)

Make this polynomial to zero
X^3 - 10x^2 + ax + b = 0. As it is divisible by the given (x - 1) & (x - 2).

CASE 1.
(x - 1) = 0
x = 1
Subsitute in X^3 - 10x^2 + ax + b = 0
(1)^3 - 10(1)^2 + a(1) + b = 0
1 - 10 + a + b = 0
- 9 + a + b = 0
a + b = 9 ________________(1)

CASE 2.
(x - 2) = 0
x = 2
Subsitute in X^3 - 10x^2 + ax + b = 0
(2)^3 - 10(2)^2 + a(2) + b = 0
8 - 40 + 2a + b = 0
- 32 + 2a + b = 0
2a + b = 32 ________________(2)

Subtract (1) from (2)
[2a + b = 32] - [a + b = 9]
2a - a + b - b = 32 - 9
a = 23

Subsitute this value in (2)
(2x 23) + b = 32
46 + b = 32
b = 32 - 46
b = - 14

Therefore, the value of a is 23
and value of b is - 14.