What is remainder theorem? Prove the remainder theorem

1.	What is remainder theorem? Prove the remainder theorem
Answer» 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).Proof : Let p(x) be any polynomial with degree greater than or equal to 1. Supposethat when p(x) is divided by x – a, the quotient is q(x) and the remainder is r(x),i.e.,p(x) = (x – a) q(x) + r(x)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. This means that r(x) is a constant, say r.So, for every value of x, r(x) = r.Therefore, p(x) = (x – a) q(x) + rIn particular, if x = a, this equation gives usp(a) = (a-a) q(a) + r = rwhich proves the theorm.

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