How To Find Rational Zeros Of Polynomials

Rational zeros of a polynomial are numbers that, when plugged into the polynomial expression, will return a zero for a result. Rational zeros are also called rational roots and x-intercepts, and are the places on a graph where the function touches the x-axis and has a zero value for the y-axis. Learning a systematic way to find the rational zeros can help you understand a polynomial function and eliminate unnecessary guesswork in solving them.

Step 1

Determine the degree of the polynomial to find the maximum number of rational zeros it can have. For example, for the polynomial x^2 – 6x + 5, the degree of the polynomial is given by the exponent of the leading expression, which is 2. The example expression has at most 2 rational zeroes.

Step 2

Find all the factors of the constant expression. For example, the constant expression in the polynomial x^2 – 6x + 5 is 5. Its factors are 1 and 5.

Step 3

Find all the factors for the leading coefficient. The leading coefficient in the polynomial equation x^2 – 6x + 5 is 1. Its only factor is 1.

Step 4

Divide the factors of the constant by the factors of the leading coefficient. For the example, the products are 1 and 5.

Step 5

Plug both the positive and negative forms of the products into the polynomial to obtain the rational zeroes. For the example, plugging 1 into the equation results in (1)^2 – 6*(1) + 5 = 1-6+5 = 0, so 1 is a rational zero.

Step 6

Continue plugging each product in to find the rational zeros. Plugging 5 into the equation results in (5)^2 – 6*(5) + 5 = 25-30+5=0, so 5 is another rational zero. Since this polynomial expression has at most 2 rational zeroes, those zeros are 1 and 5.