Question

let f(x)=3x^3-5x^2-8k-1, where k is a constant. When f(x) is divided by x-k, the quotient is g(x) and the remainder is 3k+2, find the value(s) of k

Answer 1

Let p(x) =3x³-5x²-8k-1
and g(x) =x-k
By remainder theorem
Put g(x) =0
x-k=0
x=k
So
P(k) =remainder
3k³-5k²-8k-1=3k+2
3k³-5k²-11k-3=0
Now factors of 3are 3,-3
So p(3)==81-45-33-3
=0
Hence k-3 is the factor of p(k)
So p(k) =3k³-9k²+4k²-12k+k-3
=3k²(k-3)+4k(k-3)+(k-3)
=(3k²+4k+1)(k-3)
=(3k²+3k+k+1)(k-3)
={3k(k+1)+(k+1)}(k-3)
=(k-3)(3k+1)(k+1)
To find the values
P(k) =0
So k=3,-1/3,-1