Question

Find the remainder when x⁴⁵, is divided by x² - 1.​

Answer 1

Answer:

x^2-1 =0

x^2=1

x = 1

x^45 = (1)^45

x = 1

Answer 2

Answer:

Well assuming f(x)=x^45

Let the remainder r(x)=ax +b bad quotient=q(x)

Then f(x)=x^45=q(x) ( x^2–1) + ax +b……………(1)

when f(x) is divided by x-1 then r(1)=f(1)

Or Putting x=1 in(1)

(1)^45=q(1) ( 1–1)+a+b

or a+b=1…………………………..(2)

furthermore when f(x) is divided by x+1 then r(-1)=f(-1)

Or Putting x=-1 in(1)

(-1)^45=q(-1) ( 1–1)-a+b

or -a+b=-1…………………………..(3)

Adding(2) and (3) we get

2b=0, b=0

and from (1) a=1

Thus the remainder r(x)= (1)*x+0=x