Question

explain reminder theorem with proof

Answer 1

The remainder theorem states the following: If you divide a polynomial f(x) by (x - h), then the remainder is f(h). The theorem states that ourremainder equals f(h). Therefore, we do not need to use long division, but just need to evaluate the polynomial when x = h to find the remainder.
Proof:

First convince yourself that a value nn divided by a value kk is equal to a quotient qq plus a remainder rr:

nk=q+r⟹n=q⋅k+rnk=q+r⟹n=q⋅k+r

Make some substitutions: n→f(x),q→q(x),k→(x−a)n→f(x),q→q(x),k→(x−a)

⟹f(x)=q(x)(x−a)+r⟹f(x)=q(x)(x−a)+r


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Using the polynomial remainder theorem, replacing xx with aa will yield the remainder when f(x)f(x) is divided by (x−a)(x−a).

⟹f(a)=q(a)(a−a)+r⟹f(a)=q(a)(a−a)+r

⟹f(a)=r

Answer 2

in division, down side value is remains and not get divide called remainder . ex