Factor Theorem

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Factor Theorem

Introduction to factor theorem and its uses in math contests.

Author: Jaden Park

Formula

The factor theorem states that for a polynomial $P(x),$ then $x - a$ is a factor of $P(x)$ if and only if $P(a) = 0.$

Here are some practice problems.

Given $x - 1$ is a factor of polynomial $P(x) = 2x^{3} + ax^{2} - 2x + b.$ When $P(x)$ is divided by $(x - 2),$ the remainder is $-6.$

Find the value of $a$ and $b.$

(1975 USAMO Q3)

If $P(x)$ denotes a polynomial of degree $n$ such that

P(k) = \frac{k}{k + 1}

for $k = 0, 1, 2, \ldots , n,$ determine $P(n + 1).$

Would memorize, used in school and can be useful when given a root of a polynomial.

SFFT

Sophie Germain Identity