Question

Prove Remainder theorem.

Answer 1

Remainder theorem: Let p(x) be any polynomial of degree ≥ 1, and 'a' is any real number. If p(x) is divided by (x-a), then the remainder is p(a).

Proof: Suppose q(x) is the quotient and r(x) is the remainder, when (x-a) divides p(x).

Since Dividend = Divisor $\times$ Quotient + Remainder.

Similarly we can have,

p(x) = (x-a) $\times$ q(x) + r(x) ...(I)

Where r = 0 or degree of r(x) < degree of (x -a).

Since (x-a) is a linear equation. The degree of (x-a) is 1 and degree of r(x) is less than the degree of x-a, ⇒ the degree of r(x) = 0.

It means r(x) is a constant, say r.

∴ Eqn. (I) will become:

p(x) = (x-a) $\times$ q(x) + c

Replacing x by a i.e. x =a , we have

p(a) = (a-a) $\times$ q(a) + c = 0 + c = c

⇒p(a) = c which proves the remainder theorem.