Question

if f(x) = x^4 - 2x^3 + 3x^2 - ax + b divided by (x - 1) and (x + 1), we get the same remainder 5 and 19 respectively. Find the remainder when f(x) is divided by (x - 2)

Answer 1

Answer:

We have,

p(x)=x^4–2x^3+3x^2-ax+b

By remainder theorem, when p(x) is divided by (x-1) and (x+1) , the remainders are equal to p(1) and p(-1) respectively.

By the given condition, we have

p(1)=5 and p(-1)=19

=> (1)^4–2(1)^3+3(1)^2-a(1)+b=5 and (-1)^4–2(-1)^3+3(-1)^2-a(-1)+b=19

=> 1–2+3-a+b=5 and 1-(-2)+3+a+b=19

=> -a+b=5–1+2–3 and 1+2+3+a+b=19

=> -a+b=3 and a+b=19–1–2–3

=> -a+b=3 and a+b=13

Adding these two equations,we get

-a+b+a+b=3+13

=> 2b=16

=> 2b/2=16/2

=> b=8

Putting b=8 in a+b=13 , we get

a+8=13

=> a=13–8

=> a=5

Therefore, a=5 and b=8 .