What is The Remainder theorem?
The remainder theorem is a formula that is used to find the remainder when a polynomial is divided by a linear polynomial.
What Is Remainder Theorem Formula?
p(x) = (x – a) q(x) + r(x)
Statement: Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).
Given: p(x) be any polynomial with a degree greater than or equal to 1 and p(x) is divided by x – a the quotient is q(x) and the remainder is r(x).
To Prove: the remainder is p(a).
Proof: Let p(x) be any polynomial with a degree greater than or equal to 1. Suppose that when p(x) is divided by x – a, the quotient is q(x) and the remainder is r(x), i.e.,
= Dividend = (Divisor × Quotient) + Remainder
= p(x) = (x – a) q(x) + r(x) …………..(i)
Since the degree of x – a is 1 and the degree of r(x) is less than the degree of x – a, the degree of r(x) = 0.
So, for every value of x, r(x) = r.
= p(x) = (x – a) q(x) + r ………….(ii)
In particular, if x = a, then eq (ii) becomes
p(a) = (a – a) q(a) + r
= r.