Question

8 th ques
plz solve it as soon as possible

Answer 1

Factor theorem says that if (x - c) is a factor of f(x) , then f(c) = 0 


therefore, if (x - 1) is a factor, then f(1) = 0 
1 - 10 + a + b = 0 
a + b = 9 

if (x - 2) is a factor, then f(2) = 0 
2^3 - 10(2^2) + 2a + b = 0 
8 - 40 + 2a + b = 0 
2a + b = 32 

system: 
2a + b = 32 
a + b = 9 
---------------- 
a = 23 
and if a = 23, then b = -14 

f(x) = x^3 - 10x^2 + 23x - 14 is divisible by both (x - 1) and (x -2) 


remainder theorem says that f(c) is the remainder of f(x) / (x - c) 

if x - 2 is a factor, then f(2) = 0 
2^3 + a(2^2) +2b + 6 = 0 
4a + 2b + 14 = 0 
2a + b + 7 = 0 
2a + b = -7 

if (x - 3) leaves a remainder of 3, then f(3) = 3 
3^3 + a3^2 + 3b + 6 = 3 
9a + 3b + 33 = 3 
9a + 3b = -30 
3a + b = -10 

3a + b = -10 
2a + b = -7 
----------------- 
a = -3 
and if a = -3, then b = -1 

x^3 -3x^2 - x + 6 is divisible by (x - 2), and leaves a remainder of 3 when divided by (x - 3)
Hope it helps to you.