Explain Factor Theorem.

1.	Explain Factor Theorem.
Answer» Statement: If p(x), a polynomial in x is divided by x-a and the remainder = p (a) is zero, then (x-a) is a factor of p(x). Proof: When p(x) is divided by x-a, R = p(a) (by remainder theorem) p(x) = (x-a).q(x)+p(a) (Dividend = Divisor x quotient + Remainder Division Algorithm) But p(a) = 0 is given. Hence p(x) = (x-a).q(x) => (x-a) is a factor of p(x). Conversely if x-a is a factor of p(x) then p(a) = 0. p(x) = (x-a).q(x) + R If (x-a) is a factor, then the remainder should be zero (x - a divides p(x) exactly) R = 0 By remainder theorem, R = p(a) => p(a) = 0

Answer»

Statement:

If p(x), a polynomial in x is divided by x-a and the remainder = p (a) is zero, then (x-a) is a factor of p(x).

Proof:

When p(x) is divided by x-a,

R = p(a) (by remainder theorem)

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

(Dividend = Divisor x quotient + Remainder Division Algorithm)

But p(a) = 0 is given.

Hence p(x) = (x-a).q(x)

=> (x-a) is a factor of p(x).

Conversely if x-a is a factor of p(x) then p(a) = 0.

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

If (x-a) is a factor, then the remainder should be zero (x - a divides p(x)

exactly)

R = 0

By remainder theorem, R = p(a)

=> p(a) = 0

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