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Rational Zero Theorem

If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form ± p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Example 1: Find all the rational zeros of




According to the rational zero theorem, any rational zero must have a factor of 3 in the numerator and a factor of 2 in the denominator.




The possibilities of p/q, in simplest form, are




These values can be tested by using direct substitution or by using synthetic division and finding the remainder. Synthetic division is the better method because if a zero is found, the polynomial can be written in factored form and, if possible, can be factored further, using more traditional methods.

Example 2: Find rational zeros of 2 x3 + 3 x2 − 8 x + 3 by using synthetic division.




The zeros of f ( x)=2 x3 + 3 x2 − 8 x + 3 are 1, ½, and −3. This means




The zeros could have been found without doing so much synthetic division. From the first line of the chart, 1 is seen to be a zero. This allows f( x) to be written in factored form using the synthetic division result.




But 2 x2 + 5 x − 3 can be further factored into (2 x − 1)( x + 3) using the more traditional methods of factoring.




From this completely factored form, the zeros are quickly recognized. Zeros will occur when




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