Question

what is difference between remainder therom & factor therom plz tell fast

Answer 1

According to Remainder theorm if a polynomial P(x) of degree greater than or equal to 1 is divided by a linear polynomial (x-a) then the remainder will be P(a)

e.g P(x) = x3+ 3x2+3x +1 when divided by (x+1) then as per remainder theorm the remainder is given by

P(-1)

Remainder = P(-1)

= -1 + 3 - 3 +1

= 0

Letp(x) be a polynomial of degree greater than or equal to one andabeareal number such thatp(a) = 0. Then, we have to show that (xa) is a factor ofp(x).Letq(x) be the quotient whenp(x) is divided by (x a).By remainder theoremDividend = Divisor × Quotient + Remainderp(x) = (x a) ×q(x) +p(a) [Remainder theorem]⇒p(x) = (x a) ×q(x) [p(a) = 0]⇒ (x a) is a factor ofp(x)Conversely, let (x a) be a factor ofp(x). Then we have to prove thatp(a) = 0Now(x a) is a factor ofp(x)⇒p(x), when divided by (x a) gives remainder zero. But, by the remainder theorem,p(x) when divided by (x a) gives the remainder equal top(a).∴p(a) = 0the difference might be that in factor thm. we have to guess the zero of the polynomial.Hit the thumbs up if u like the ans.

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