How to Find Rational Zeros of Polynomials

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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.

    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.

    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.

    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.

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

    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.


    • This method of finding the rational zeroes works with any degree of polynomial.

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