# Remaining theorem of polynomial if divided by x-k

Suppose the value of the polynomial f(k) = 0, then the
polynomial f(x) remains 0 if divided by

**x - k**. Thus,**(x - k)**is said to be a factor of f(x).
Polynomial factor theorem

The polynomial f(x) has a factor (x - k) if and only
if f(k) = 0

Example 1.

Show that x - 3 is a factor of the polynomial f(x) = x

^{3}– x^{2}– x – 15
Answers:

Apparently, the result is f(x) = 0, so x - 3 is a factor of f(x) or x

^{3}– x^{2}– x – 15 = (x -3)(x^{2}+ 2x + 5).
Example 2.

Solve the polynomial equation x

^{3}– 11x^{2}+ 30x – 8 by factoring on**rational factors**.
Answer:

For example, one of these factors is x - k, then k is a factor of (- 8). Possible factors of (- 8) are ± 1, ± 2, ± 4 and ± 8.

By experimenting with some of the numbers above, we find the remainder of the division 0 for x = 4, that is

So,

**the rational factor**is x - 4.SUBSCRIBE TO OUR NEWSLETTER

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