Question

the remainder when x²⁰¹⁴-1 is divided by x-1 is​

Answer 1

Answer:

When a polynomial f(x) is divided by (x-1) and (x-2), it leaves remainder 5 and 7 respectively. What is the remainder when divided by (x-1) (x-2)?

Let us assume the polynomial to be f(x) .

When f(x) is divided by (x−1) , we get the remainder as 5 .

Therefore, f(1)=5

When f(x) is divided by (x−2) , we get the remainder as 7 .

Therefore, f(2)=7

Now, when same polynomial f(x)

is divided by (x−1)(x−2) , the remainder is given by:

[math]f\left( x \right) = q\[/math]

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F(x) = (x-1) q(x) + 5 …..1

F(x) = (x-2) q'(x) + 7 ……2

So F(1) = 5 and F(2) = 7.

Now F(x) is divided by (x-1)(x-2) it's is a polynomial of degree two.

Reminder should be a polynomial of degree less then 2. Say.. r=ax+b

Now.. F(x) = (x-1)(x-2) q”(x) + r

F(x) = (x-1)(x-2) q”(x) + (ax+b). ………3

@x=1. 5= a+b

@x=2. 7= 2a+b

Answer 2

remainder will be 0

x - 1 = 0

x = 1

then,, ( 1 ) ^ 2014 - 1 = 0