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Polynomials: The Rational Zero Test

The Rational Zero Test states that if a polynomial with real coefficients has rational zeros each will be on the following list: c/d, where c is a factor of the constant term and d is a factor of the leading coefficient.

Example: Make a list of possible rational zeros of the polynomial, P(x) = 4x3 + 19x2 + 20x + 6.

The constant term is 6. Its factors are 1, 2, 3 and 6. These are values of c.

The leading coefficient is 4. Its factors are 1, 2, and 4. These are values of d.

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Polynomials: The Rational Zero Test

Slide 2

c-values – 1, 2, 3 and 6 d-values – 1, 2 and 4

Now form all fractions of the form: c/d:

.46

,43

,42

,41

,26

,23

,22

,21

,16

,13

,12

,11

.43

,41

,23

,21

,6,3,2,1

Simplifying and removing duplicate fractions results in:

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Polynomials: The Rational Zero Test

Slide 3

The polynomial in the example,(P(x) = 4x3 + 19x2 + 20x + 6) actually has three real zeros: .22,22,

43

Two of the zeros are irrational; only one is rational (- 3/4). The Rational Zero Test makes no claims beyond those for rational zeros. It simply states that if any rational zeros exist, they will appear on the computed list. Indeed, - 3/4 was on the computed list!

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Polynomials: The Rational Zero Test

Slide 4

Try: Make a list of possible rational zeros of the polynomial, P(x) = 5x3 – 14x2 + 3x + 4.

The possible rational zeros are: .54

,52

,51

,4,2,1

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Polynomials: The Rational Zero Test