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.
If is a factor of , then , where is a polynomial with . Then .
Now suppose that .
Substitute and get . Since is a constant polynomial, for all .
Therefore, , which shows that is a factor of .
Here are some problems that can be solved using the Factor Theorem:
Suppose is a -degrees 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)
If denotes a polynomial of degree such thatfor , determine .
(Source: 1975 USAMO Problem 3)
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