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Zeros of Polynomial Functions (Part 1)

# Zeros of Polynomial Functions (Part 1) - Rational Zero(or...

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Rational Zero (or Root) Theorem If , where are integer coefficients and the reduced fraction is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient . We can use this theorem to help us find all of the POSSIBLE rational zeros or roots of a polynomial function. Step 1: List all of the factors of the constant. In the Rational Zero Theorem, p represents factors of the constant term. Make sure that you include both the positive and negative factors. Step 2: List all of the factors of the leading coefficient. In the Rational Zero Theorem, q represents factors of the leading coefficient. Make sure that you include both the positive and negative factors. Step 3: List all the POSSIBLE rational zeros or roots. This list comes from taking all the factors of the constant ( p ) and writing

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them over all the factors of the leading coefficient ( q ), to get a list of . Make sure that you get ever possible combination of these factors, written as . Example 1 : Use the Rational Zero Theorem to list all the possible rational zeros for . Step 1: List all of the factors of the constant. The factors of the constant term 12 are . Step 2: List all of the factors of the leading coefficient. The factors of the leading coefficient -1 are . Step 3: List all the POSSIBLE rational zeros or roots.
Writing the possible factors as we get: Generally we don’t write a number over 1, I just did it to emphasize that the denominator comes for factors of q which are . It would have been ok to write out the 2nd line without writing out the 1st line. Example 2 : Use the Rational Zero Theorem to list all the possible rational zeros for . Step 1: List all of the factors of the constant. The factors of the constant term -20 are . Step 2: List all of the factors of the leading coefficient.

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The factors of the leading coefficient 6 are . Step 3: List all the POSSIBLE rational zeros or roots. Writing the possible factors as we get: Note, how when you reduce down the fractions, some of them are repeated. Here is a final list of all the POSSIBLE rational zeros, each one written once and reduced: Example 3 : List all of the possible zeros, use synthetic division to test the possible zeros, find an actual zero, and use the actual zero to find all the zeros of .
List all of the possible zeros: The factors of the constant term 100 are . The factors of the leading coefficient 1 are . Writing the possible factors as we get: Use synthetic division to test the possible zeros and find an actual zero: Recall that if you apply synthetic division and the remainder is 0, then c is a zero or root of the polynomial function. If you need a review on synthetic division, feel free to go to Tutorial 37: Synthetic Division and the Remainder and Factor Theorems. At this point you are wanting to pick any POSSIBLE rational root form the list of

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Zeros of Polynomial Functions (Part 1) - Rational Zero(or...

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