Factor Theorem

 

Definition: 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 (x). Thus, the theorem states that if f (c)=0, then (xc) 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 theorem:

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 – cq(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.

The Remainder theorem says, if (x - c) divides the polynomial f (x), then the remainder is f (c) That is,

f (x) = (x – cq(x) + f (c)

Suppose the remainder f (c) = 0, f (x) = (x – cq(x).

Thus, (x – c) is the factor of f (x). Hence, it can be concluded that the “Factor theorem” is the reverse of “Remainder theorem”.

Example:

Consider a polynomial . Determine whether (x+1) is a factor of f (x).

By the Factor theorem, (x + 1) is a factor of f (x) if f (1) = 0.

Obtain the value of f (1).

Since f (1) = 0, (x + 1) is a factor of f (x).