Question

state and prove factor theorem



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Answer 1

FACTOR THEORAM

Let f (x) be a polynomial. If a polynomial f (x) is divided by x = c, then the remainder will be zero. That is, x = c is zero or root of a polynomial f (x) , which also makes (x – c) is a factor of f (x). Thus, the theorem states that if f (c)=0, then (x–c) is a factor of the polynomial f (x). The converse of this theorem is also true. That is, if (x – c) is a factor of the polynomial f (x), then f(c)=0.

PROOF OF FACTOR THEORAM

Consider a polynomial f (x) which is divided by (x – c) .

Then, f (c) = 0.

Thus, by the Remainder theorem,

Thus, (x – c) is a factor of the polynomial f (x).

Proof of the converse part:

By the Remainder theorem,

f (x) = (x – c) q(x) + f (c)

If (x – c) is a factor of f (x), then the remainder must be zero.

That is, (x – c) exactly divides f (x).

Thus, f (c) = 0.

Hence proved.

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