Explain Remainder Theorem.

1.	Explain Remainder Theorem.
Answer» If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a). Proof: Let p(x) be a polynomial divided by (x-a). Let q(x) be the quotient and R be the remainder. By division algorithm, Dividend = (Divisor x quotient) + Remainder p(x) = q(x) . (x-a) + R Substitute x = a, p(a) = q(a) (a-a) + R p(a) = R (a - a = 0, 0 - q (a) = 0) Hence Remainder = p(a). Steps for Factorization using Remainder Theorem By trial and error method, find the factor of the constant for which the given expression becomes equal to zero. Divide the expression by the factor that is determined in step 1. Factorize the quotient. If the quotient is a trinomial, factorize it further. If the expression is a 4^th degree expression, the first step will be to reduce this to a trinomial and then factorize this trinomial further.

Answer»

If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a).

Proof:

Let p(x) be a polynomial divided by (x-a).

Let q(x) be the quotient and R be the remainder.

By division algorithm,

Dividend = (Divisor x quotient) + Remainder

p(x) = q(x) . (x-a) + R

Substitute x = a,

p(a) = q(a) (a-a) + R

p(a) = R (a - a = 0, 0 - q (a) = 0)

Hence Remainder = p(a).

Steps for Factorization using Remainder Theorem

By trial and error method, find the factor of the constant for which the given expression becomes equal to zero.
Divide the expression by the factor that is determined in step 1.
Factorize the quotient. If the quotient is a trinomial, factorize it further.
If the expression is a 4^th degree expression, the first step will be to reduce this to a trinomial and then factorize this trinomial further.

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