Question

What is the factor theorem?exaplan

Answer 1

Let p(x) be a polynomial of a degree greater than or equal to one and a be a real number such that p(a) = 0

Then , we have to show that ( x-a ) is a factor of p(x).

Let q(x) be the quotient when p(x) is divisible by ( x-a)

By remainder theorem

Dividend = Divisor x Quotient + Remainder  

p(x) = ( x-a ) x q(x) + p(a) [Remainder theorem]

p(x) = (x-a) x q(x)  [p(a) = 0 ]

(x-a) is a factor of p(x)

conversely ,  

Let (x-a) be a factor of p(x) . Then we have to prove that p(a) = 0

Now  

(x-a) is a factor of p(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 to p(a)

.: p(a) = 0

Hope You Understand  

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