The Remainder Theorem states that if dividing a polynomialby leaves a remainder of , then .
The Rational Zero Theorem gives us a number of options for the rational zeros of a polynomial. All of the polynomial's rational zeros will be contained in the set the theorem tells us, but not all of the members of the set will be actual zeros of the polynomial. The theorem states that if we have some polynomial, where is the leading term, then we can find its rational zeros by dividing a factor of by a factor of . The Rational Zero Theorem narrows down the possibilities for what the zeros of our polynomial could be. As an example, let's say that , then and . The factors of are , , , and , and the factors of are , , , , , , , and . Dividing each factor of by each factor of gives us a set of numbers, which we'll write a bit more compactly using signs: , , , , , and . The actual rational zeros of are , , and , all of which the theorem told us were possibilities.