Question

State and prove 'Remainder theorai​

Answer 1

Statement of Remainder Theorem:

Let f(x) be any polynomial of degree greater than or equal to one and let ‘ a‘ be any number. If f(x) is divided by the linear polynomial (x-a) then the remainder is f(a).

Remainder Theorem Proof:

Let f(x) be any polynomial with degree greater than or equal to 1.

Further suppose that when f(x) is divided by a linear polynomial p(x) = ( x -a), the quotient is q(x) and the remainder is r(x).

In other words , f(x) and p(x) are two polynomials such that the degree of f(x) \geq degree of p(x) and p(x) \neq 0 then we can find polynomials q(x) and r(x) such that, where r(x) = 0 or degree of r(x) < degree of g(x).

By division algorithm

f(x) = p(x) . q(x) + r(x)

∴ f(x) = (x-a) . q(x) + r(x) [ here p(x) = x – a ]

Since degree of p(x) = (x-a) is 1 and degree of r(x) < degree of (x-a)

∴ Degree of r(x) = 0

This implies that r(x) is a constant , say ‘ k ‘

So, for every real value of x, r(x) = k.

Therefore f(x) = ( x-a) . q(x) + k

If x = a,

then f(a) = (a-a) . q(a) + k = 0 + k = k

Hence the remainder when f(x) is divided by the linear polynomial (x-a) is f(a).

Statement of Factor Theorem:

If f(x) is a polynomial of degree n \geq 1 and ‘ a ‘ is any real number then

1. (x -a) is a factor of f(x), if f(a) = 0.

2. and its converse ” if (x-a) is a factor of a polynomial f(x) then f(a) = 0 “

Factor Theorem Proof:

Given that f(x) is a polynomial of degree n \geq 1 by reminder theorem.

f(x) = ( x-a) . q(x) + f(a) . . . . . . . . . . . \rightarrow equation ‘A ‘

1 . Suppose f(a) = 0

then equation ‘A’ \Rightarrow f(x) = ( x-a) . q(x) + 0 = ( x-a) . q(x)

Which shows that ( x-a) is a factor of f(x). Hence proved

2 . Conversely suppose that (x-a) is a factor of f(x).

This implies that f(x) = ( x-a) . q(x) for some polynomial q(x).

∴ f(a) = ( a-a) . q(a) = 0.

Hence f(a) = 0 when (x-a) is a factor of f(x).

The factor theorem simply say that If a polynomial f(x) is divided by p(x) leaves remainder zero then p(x) is factor of f(x)

Answer 2

Answer:

The remainder theorem states that when a polynomial is f (a) is divided by another binomial (a-x) then the remainder of the end result that is obtained by f (a) . The remainder theorem is applicable only when the polynomial can be decided entirely atleast one time by the binomial factor to reduce the bigger polynomial a and the remainder to be 0

when f(a) is divided by (a-x) , then :

f (a) = ( a - x ) . q (a ) + r

consider x = a ;

Then ,

f ( a ) = ( a - ) . q ( a ) + r

f ( a ) = r