Question

x^3-10x^2+ax+b, find a and b so the equation is exactly divisible by (x-2),and (x-1)

Answer 1

Let p(x) = x3 - 10x2 +ax +b Using factor theorem we find that if p(x) is divided by x-1 the remainder, i.e. p(1) = 0

Therefore, p(1) = (1)3 - 10(1)2 + a(1) +b = 0

=> 1 - 10 +a +b =0

=> a+ b = 9=> b = 9-a .........equation (1) 
In the same way if p(x) is divided by x-2 the remainder, i.e. p(2) = 0 Therefore, p(2) = (2)3 - 10(2)2 + a(2) +b = 0

=> 8 - 40 +2a +b =0=> -32+2a+b=0...........equation (2)

Substituting the value of equation(1) in equation(2) we get, -32 + 2a+9 -a = 0

=> a = 23

Substituting the value of a in equation (1) we get b= 9- 23

=> b = -14

Hence, a=23 and b=-14h