Question

Find numbers a and k so that x-2 is a factor of f(x) = x^4-2ax^3 +ax^2-x+k and f(-1)=3

Answer 1

Answer:

a = 1, k = -2

Step-by-step explanation:

f(x) = x^4-2ax^3 +ax^2-x+k

The remainder should be 0 when f(x) is divided by (x-2)

or x-2=0

x=2

∴f(2) = 0 = 2^4-2a2^3 +a2^2-2+k

16-16a+4a-2+k=0

14-12a+k=0

12a - k = 14 ----- eqn 1

Using remainder theorem,

f(-1) = 3 = (-1)^4-2a(-1)^3 +a(-1)^2-(-1)+k

1+2a+a+1+k=3

3a + k = 1   ----- eqn 2

Adding eqn 1&2, we get

15a = 15

a = 1

Putting this value of a in eqn 1, we get

12(1) - k = 14

k = -2