Question

if f (x) is a polynomial in x of degree >1 and "a" is any real number and if f (x) is divided by (x-a),then the remainder is ?

Answer 1

Answer:

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Answer 2

Answer:

The Remainder Theorem for polynomials states that the remainder obtained by dividing a polynomial f(x)f(x) by (x−a)(x−a) is f(a)f(a).

Proof:

f(x)=g(x)⋅q(x)+r(x)f(x)=g(x)⋅q(x)+r(x)

Here, g(x)=(x−a)g(x)=(x−a)

⟹f(x)=(x−a)⋅q(x)+r(x)⟹f(x)=(x−a)⋅q(x)+r(x)

But, deg[r(x)]<deg[g(x)]deg[r(x)]<deg[g(x)]

Hence, the degree of r(x)r(x) is 00.

Hence, r(x)r(x) is a constant.

Let r(x)=rr(x)=r for any real number rr.

⟹f(x)=(x−a)⋅q(x)+r⟹f(x)=(x−a)⋅q(x)+r

Replacing xx by aa, we get

f(a)=(a−a)⋅q(a)+rf(a)=(a−a)⋅q(a)+r

⟹f(a)=r