Glossary for Lesson 21: Rational Zero Test

Remainder Theorem

The Remainder Theorem states that if dividing a polynomial p(x) by (x-k) leaves a remainder of r , then p(k)=r .

Rational Zero Theorem

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 f(x)=px^n+...+q , where px^n is the leading term, then we can find its rational zeros by dividing a factor of q by a factor of p. The Rational Zero Theorem narrows down the possibilities for what the zeros of our polynomial could be. As an example, let's say that f(x)=2x^3-5x^2-x+6 , then p=2 and q=6 . The factors of 2 are 1 , 2 , -1 , and -2 , and the factors of 6 are 1 , 2 , 3 , 6 , -1 , -2 , -3 , and -6 . Dividing each factor of q by each factor of p gives us a set of numbers, which we'll write a bit more compactly using \pm signs: \pm 1 , \pm 2 , \pm 3 , \pm 6 , \pm\frac{1}{2} , and \pm\frac{3}{2} . The actual rational zeros of f(x)=2x^3-5x^2-x+6 are \frac {3}{2} , -1 , and 2 , all of which the theorem told us were possibilities.