# Factor Theorem

In algebra, the **Factor theorem** is a theorem regarding the relationships between the factors of a polynomial and its roots.

One of it's most important applications is if you are given that a polynomial have certain roots, you will know certain linear factors of the polynomial. Thus, you can test if a linear factor is a factor of a polynomial without using polynomial division and instead plugging in numbers. Conversely, you can determine whether a number in the form ( is constant, is polynomial) is using polynomial division rather than plugging in large values.

## Statement

The **Factor Theorem** says that if is a polynomial, then is a factor of if and only if .

## Proof

If is a factor of , then , where is a polynomial with . Then .

Now suppose that .

Apply Remainder Theorem to get , where is a polynomial with and is the remainder polynomial such that . This means that can be at most a constant polynomial.

Substitute and get . Since is a constant polynomial, for all .

Therefore, , which shows that is a factor of .

## Problems

Here are some problems that can be solved using the Factor Theorem:

### Introductory

### Intermediate

Suppose is a -degree polynomial. The Fundamental Theorem of Algebra tells us that there are roots, say . Suppose all integers ranging from to satisfies . Also, suppose that

for an integer . If is the minimum possible positive integral value of

.

Find the number of factors of the prime in . (Source: I made it. Solution here)

### Olympaid

If denotes a polynomial of degree such thatfor , determine .

(Source: 1975 USAMO Problem 3)

## See Also

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