Question

do anybody wants to see the proof for remainder theorem & factor theorem:)

Answer 1

Using the division lemma . we have,\[f(x)=(x-a)q(x)+r\]Now,if we put x=a, we get:\[f(a)=(a-a)q(a)+r=0.q(a)+r=r\]so,the remainder theorem when f(x) is divided by x-a is f(a). this fact is known as remainder theorem.

Answer 2

suppose that (x-a) divides f(x) then r=0 & we get:\[f(x)=(x-a).q(x)\]hence , f(a)=0

Answer 3

factor theorem:
to prove ,this divide f(x) by (x-a) to gt q(x) & r such that\[f(x)=(x-a).q(x)+r\]now if f(a)=0,then putting x=a,we gt:\[0=f(a)=a-a.q(a)+r\]\[=0+r\]or r=0